Let $p_1,p_2,p_3$ are consecutive prime s.t. $p_3>p_2>p_1>3$
Then show that
If $(\frac{p_2-p_1}{2})\equiv1\pmod3$ then $ (\frac{p_3-p_2}{2})\not\equiv1\pmod3$
If $(\frac{p_2-p_1}{2})\equiv2\pmod3$ then $ (\frac{p_3-p_2}{2})\not\equiv2\pmod3$
Source code
? forprime(p=4, 10000, print ([p,((nextprime(p+1)-p)/2)%3]))
[5, 1]
[7, 2]
[11, 1]
[13, 2]
[17, 1]
[19, 2]
[23, 0]
[29, 1]
[31, 0]
[37, 2]
[41, 1]
[43, 2]
[47, 0]
[53, 0]
[59, 1]
[61, 0]
[67, 2]
[71, 1]
[73, 0]
[79, 2]
[83, 0]
[89, 1]
[97, 2]
[101, 1]
[103, 2]
[107, 1]
[109, 2]
[113, 1]
[127, 2]
[131, 0]
[137, 1]
[139, 2]
[149, 1]
[151, 0]
[157, 0]
[163, 2]
[167, 0]
[173, 0]
[179, 1]
[181, 2]
[191, 1]
[193, 2]
[197, 1]
[199, 0]
[211, 0]
[223, 2]
[227, 1]
[229, 2]
[233, 0]
[239, 1]
[241, 2]
[251, 0]
[257, 0]
[263, 0]
[269, 1]
[271, 0]
[277, 2]
[281, 1]
[283, 2]
[293, 1]
[307, 2]
[311, 1]
[313, 2]
[317, 1]
[331, 0]
[337, 2]
[347, 1]
[349, 2]
[353, 0]
[359, 1]
[367, 0]
[373, 0]
[379, 2]
[383, 0]
[389, 1]
[397, 2]
[401, 1]
[409, 2]
[419, 1]
[421, 2]
[431, 1]
[433, 0]
[439, 2]
[443, 0]
[449, 1]
[457, 2]
[461, 1]
[463, 2]
[467, 0]
[479, 1]
[487, 2]
[491, 1]
[499, 2]
[503, 0]
[509, 0]
[521, 1]
[523, 0]
[541, 0]
[547, 2]
[557, 0]
[563, 0]
[569, 1]
[571, 0]
[577, 2]
[587, 0]
[593, 0]
[599, 1]
[601, 0]
[607, 0]
[613, 2]
[617, 1]
[619, 0]
[631, 2]
[641, 1]
[643, 2]
[647, 0]
[653, 0]
[659, 1]
[661, 0]
[673, 2]
[677, 0]
[683, 1]
[691, 2]
[701, 1]
[709, 2]
[719, 1]
[727, 0]
[733, 0]
[739, 2]
[743, 1]
[751, 0]
[757, 2]
[761, 1]
[769, 2]
[773, 1]
[787, 2]
[797, 0]
[809, 1]
[811, 2]
[821, 1]
[823, 2]
[827, 1]
[829, 2]
[839, 1]
[853, 2]
[857, 1]
[859, 2]
[863, 1]
[877, 2]
[881, 1]
[883, 2]
[887, 1]
[907, 2]
[911, 1]
[919, 2]
[929, 1]
[937, 2]
[941, 0]
[947, 0]
[953, 1]
[967, 2]
[971, 0]
[977, 0]
[983, 1]
[991, 0]
[997, 0]
[1009, 2]
[1013, 0]
[1019, 1]
[1021, 2]
[1031, 1]
[1033, 0]
[1039, 2]
[1049, 1]
[1051, 2]
[1061, 1]
[1063, 0]
[1069, 0]
[1087, 2]
[1091, 1]
[1093, 2]
[1097, 0]
[1103, 0]
[1109, 1]
[1117, 0]
[1123, 0]
[1129, 2]
[1151, 1]
[1153, 2]
[1163, 1]
[1171, 2]
[1181, 0]
[1187, 0]
[1193, 1]
[1201, 0]
[1213, 2]
[1217, 0]
[1223, 0]
[1229, 1]
[1231, 0]
[1237, 0]
[1249, 2]
[1259, 0]
[1277, 1]
[1279, 2]
[1283, 0]
[1289, 1]
[1291, 0]
[1297, 2]
[1301, 1]
[1303, 2]
[1307, 0]
[1319, 1]
[1321, 0]
[1327, 2]
[1361, 0]
[1367, 0]
[1373, 1]
[1381, 0]
[1399, 2]
[1409, 1]
[1423, 2]
[1427, 1]
[1429, 2]
[1433, 0]
[1439, 1]
[1447, 2]
[1451, 1]
[1453, 0]
[1459, 0]
[1471, 2]
[1481, 1]
[1483, 2]
[1487, 1]
[1489, 2]
[1493, 0]
[1499, 0]
[1511, 0]
[1523, 1]
[1531, 0]
[1543, 0]
[1549, 2]
[1553, 0]
[1559, 1]
[1567, 2]
[1571, 1]
[1579, 2]
[1583, 1]
[1597, 2]
[1601, 0]
[1607, 1]
[1609, 2]
[1613, 0]
[1619, 1]
[1621, 0]
[1627, 2]
[1637, 1]
[1657, 0]
[1663, 2]
[1667, 1]
[1669, 0]
[1693, 2]
[1697, 1]
[1699, 2]
[1709, 0]
[1721, 1]
[1723, 2]
[1733, 1]
[1741, 0]
[1747, 0]
[1753, 0]
[1759, 0]
[1777, 0]
[1783, 2]
[1787, 1]
[1789, 0]
[1801, 2]
[1811, 0]
[1823, 1]
[1831, 2]
[1847, 1]
[1861, 0]
[1867, 2]
[1871, 1]
[1873, 2]
[1877, 1]
[1879, 2]
[1889, 0]
[1901, 0]
[1907, 0]
[1913, 0]
[1931, 1]
[1933, 2]
[1949, 1]
[1951, 2]
[1973, 0]
[1979, 1]
[1987, 0]
[1993, 2]
[1997, 1]
[1999, 2]
[2003, 1]
[2011, 0]
[2017, 2]
[2027, 1]
[2029, 2]
[2039, 1]
[2053, 2]
[2063, 0]
[2069, 0]
[2081, 1]
[2083, 2]
[2087, 1]
[2089, 2]
[2099, 0]
[2111, 1]
[2113, 2]
[2129, 1]
[2131, 0]
[2137, 2]
[2141, 1]
[2143, 2]
[2153, 1]
[2161, 0]
[2179, 0]
[2203, 2]
[2207, 0]
[2213, 1]
[2221, 2]
[2237, 1]
[2239, 2]
[2243, 1]
[2251, 2]
[2267, 1]
[2269, 2]
[2273, 1]
[2281, 0]
[2287, 0]
[2293, 2]
[2297, 0]
[2309, 1]
[2311, 2]
[2333, 0]
[2339, 1]
[2341, 0]
[2347, 2]
[2351, 0]
[2357, 1]
[2371, 0]
[2377, 2]
[2381, 1]
[2383, 0]
[2389, 2]
[2393, 0]
[2399, 0]
[2411, 0]
[2417, 0]
[2423, 1]
[2437, 2]
[2441, 0]
[2447, 0]
[2459, 1]
[2467, 0]
[2473, 2]
[2477, 1]
[2503, 0]
[2521, 2]
[2531, 1]
[2539, 2]
[2543, 0]
[2549, 1]
[2551, 0]
[2557, 2]
[2579, 0]
[2591, 1]
[2593, 2]
[2609, 1]
[2617, 2]
[2621, 0]
[2633, 1]
[2647, 2]
[2657, 1]
[2659, 2]
[2663, 1]
[2671, 0]
[2677, 0]
[2683, 2]
[2687, 1]
[2689, 2]
[2693, 0]
[2699, 1]
[2707, 2]
[2711, 1]
[2713, 0]
[2719, 2]
[2729, 1]
[2731, 2]
[2741, 1]
[2749, 2]
[2753, 1]
[2767, 2]
[2777, 0]
[2789, 1]
[2791, 0]
[2797, 2]
[2801, 1]
[2803, 2]
[2819, 1]
[2833, 2]
[2837, 0]
[2843, 1]
[2851, 0]
[2857, 2]
[2861, 0]
[2879, 1]
[2887, 2]
[2897, 0]
[2903, 0]
[2909, 1]
[2917, 2]
[2927, 0]
[2939, 1]
[2953, 2]
[2957, 0]
[2963, 0]
[2969, 1]
[2971, 2]
[2999, 1]
[3001, 2]
[3011, 1]
[3019, 2]
[3023, 1]
[3037, 2]
[3041, 1]
[3049, 0]
[3061, 0]
[3067, 0]
[3079, 2]
[3083, 0]
[3089, 1]
[3109, 2]
[3119, 1]
[3121, 2]
[3137, 1]
[3163, 2]
[3167, 1]
[3169, 0]
[3181, 0]
[3187, 2]
[3191, 0]
[3203, 0]
[3209, 1]
[3217, 2]
[3221, 1]
[3229, 2]
[3251, 1]
[3253, 2]
[3257, 1]
[3259, 0]
[3271, 2]
[3299, 1]
[3301, 0]
[3307, 0]
[3313, 0]
[3319, 2]
[3323, 0]
[3329, 1]
[3331, 0]
[3343, 2]
[3347, 0]
[3359, 1]
[3361, 2]
[3371, 1]
[3373, 2]
[3389, 1]
[3391, 2]
[3407, 0]
[3413, 1]
[3433, 2]
[3449, 1]
[3457, 2]
[3461, 1]
[3463, 2]
[3467, 1]
[3469, 2]
[3491, 1]
[3499, 0]
[3511, 0]
[3517, 2]
[3527, 1]
[3529, 2]
[3533, 0]
[3539, 1]
[3541, 0]
[3547, 2]
[3557, 1]
[3559, 0]
[3571, 2]
[3581, 1]
[3583, 2]
[3593, 1]
[3607, 0]
[3613, 2]
[3617, 0]
[3623, 1]
[3631, 0]
[3637, 0]
[3643, 2]
[3659, 0]
[3671, 1]
[3673, 2]
[3677, 1]
[3691, 0]
[3697, 2]
[3701, 1]
[3709, 2]
[3719, 1]
[3727, 0]
[3733, 0]
[3739, 2]
[3761, 0]
[3767, 1]
[3769, 2]
[3779, 1]
[3793, 2]
[3797, 0]
[3803, 0]
[3821, 1]
[3823, 2]
[3833, 1]
[3847, 2]
[3851, 1]
[3853, 2]
[3863, 1]
[3877, 2]
[3881, 1]
[3889, 0]
[3907, 2]
[3911, 0]
[3917, 1]
[3919, 2]
[3923, 0]
[3929, 1]
[3931, 0]
[3943, 2]
[3947, 1]
[3967, 2]
[3989, 0]
[4001, 1]
[4003, 2]
[4007, 0]
[4013, 0]
[4019, 1]
[4021, 0]
[4027, 2]
[4049, 1]
[4051, 0]
[4057, 2]
[4073, 0]
[4079, 0]
[4091, 1]
[4093, 0]
[4099, 0]
[4111, 2]
[4127, 1]
[4129, 2]
[4133, 0]
[4139, 1]
[4153, 2]
[4157, 1]
[4159, 0]
[4177, 0]
[4201, 2]
[4211, 0]
[4217, 1]
[4219, 2]
[4229, 1]
[4231, 2]
[4241, 1]
[4243, 2]
[4253, 0]
[4259, 1]
[4261, 2]
[4271, 1]
[4273, 2]
[4283, 0]
[4289, 1]
[4297, 0]
[4327, 2]
[4337, 1]
[4339, 2]
[4349, 1]
[4357, 0]
[4363, 2]
[4373, 0]
[4391, 0]
[4397, 0]
[4409, 0]
[4421, 1]
[4423, 0]
[4441, 0]
[4447, 2]
[4451, 0]
[4457, 0]
[4463, 0]
[4481, 1]
[4483, 2]
[4493, 1]
[4507, 0]
[4513, 2]
[4517, 1]
[4519, 2]
[4523, 0]
[4547, 1]
[4549, 0]
[4561, 0]
[4567, 2]
[4583, 1]
[4591, 0]
[4597, 0]
[4603, 0]
[4621, 2]
[4637, 1]
[4639, 2]
[4643, 0]
[4649, 1]
[4651, 0]
[4657, 0]
[4663, 2]
[4673, 0]
[4679, 0]
[4691, 0]
[4703, 0]
[4721, 1]
[4723, 0]
[4729, 2]
[4733, 0]
[4751, 1]
[4759, 0]
[4783, 2]
[4787, 1]
[4789, 2]
[4793, 0]
[4799, 1]
[4801, 0]
[4813, 2]
[4817, 1]
[4831, 0]
[4861, 2]
[4871, 0]
[4877, 0]
[4889, 1]
[4903, 0]
[4909, 2]
[4919, 0]
[4931, 1]
[4933, 2]
[4937, 0]
[4943, 1]
[4951, 0]
[4957, 2]
[4967, 1]
[4969, 2]
[4973, 1]
[4987, 0]
[4993, 0]
[4999, 2]
[5003, 0]
[5009, 1]
[5011, 2]
[5021, 1]
[5023, 2]
[5039, 0]
[5051, 1]
[5059, 0]
[5077, 2]
[5081, 0]
[5087, 0]
[5099, 1]
[5101, 0]
[5107, 0]
[5113, 0]
[5119, 2]
[5147, 0]
[5153, 1]
[5167, 2]
[5171, 1]
[5179, 2]
[5189, 1]
[5197, 0]
[5209, 0]
[5227, 2]
[5231, 1]
[5233, 2]
[5237, 0]
[5261, 0]
[5273, 0]
[5279, 1]
[5281, 2]
[5297, 0]
[5303, 0]
[5309, 1]
[5323, 2]
[5333, 1]
[5347, 2]
[5351, 0]
[5381, 0]
[5387, 0]
[5393, 0]
[5399, 1]
[5407, 0]
[5413, 2]
[5417, 1]
[5419, 0]
[5431, 0]
[5437, 2]
[5441, 1]
[5443, 0]
[5449, 2]
[5471, 0]
[5477, 1]
[5479, 2]
[5483, 0]
[5501, 1]
[5503, 2]
[5507, 0]
[5519, 1]
[5521, 0]
[5527, 2]
[5531, 1]
[5557, 0]
[5563, 0]
[5569, 2]
[5573, 1]
[5581, 2]
[5591, 1]
[5623, 2]
[5639, 1]
[5641, 0]
[5647, 2]
[5651, 1]
[5653, 2]
[5657, 1]
[5659, 2]
[5669, 1]
[5683, 0]
[5689, 2]
[5693, 1]
[5701, 2]
[5711, 0]
[5717, 1]
[5737, 2]
[5741, 1]
[5743, 0]
[5749, 0]
[5779, 2]
[5783, 1]
[5791, 2]
[5801, 0]
[5807, 0]
[5813, 1]
[5821, 0]
[5827, 0]
[5839, 2]
[5843, 0]
[5849, 1]
[5851, 0]
[5857, 2]
[5861, 0]
[5867, 1]
[5869, 2]
[5879, 1]
[5881, 2]
[5897, 0]
[5903, 1]
[5923, 2]
[5927, 0]
[5939, 1]
[5953, 2]
[5981, 0]
[5987, 1]
[6007, 2]
[6011, 0]
[6029, 1]
[6037, 0]
[6043, 2]
[6047, 0]
[6053, 1]
[6067, 0]
[6073, 0]
[6079, 2]
[6089, 1]
[6091, 2]
[6101, 0]
[6113, 1]
[6121, 2]
[6131, 1]
[6133, 2]
[6143, 1]
[6151, 0]
[6163, 2]
[6173, 0]
[6197, 1]
[6199, 2]
[6203, 1]
[6211, 0]
[6217, 2]
[6221, 1]
[6229, 0]
[6247, 2]
[6257, 0]
[6263, 0]
[6269, 1]
[6271, 0]
[6277, 2]
[6287, 0]
[6299, 1]
[6301, 2]
[6311, 0]
[6317, 0]
[6323, 0]
[6329, 1]
[6337, 0]
[6343, 2]
[6353, 0]
[6359, 1]
[6361, 0]
[6367, 0]
[6373, 0]
[6379, 2]
[6389, 1]
[6397, 0]
[6421, 0]
[6427, 2]
[6449, 1]
[6451, 0]
[6469, 2]
[6473, 1]
[6481, 2]
[6491, 0]
[6521, 1]
[6529, 0]
[6547, 2]
[6551, 1]
[6553, 2]
[6563, 0]
[6569, 1]
[6571, 0]
[6577, 2]
[6581, 0]
[6599, 1]
[6607, 0]
[6619, 0]
[6637, 2]
[6653, 0]
[6659, 1]
[6661, 0]
[6673, 0]
[6679, 2]
[6689, 1]
[6691, 2]
[6701, 1]
[6703, 0]
[6709, 2]
[6719, 1]
[6733, 2]
[6737, 0]
[6761, 1]
[6763, 2]
[6779, 1]
[6781, 2]
[6791, 1]
[6793, 2]
[6803, 1]
[6823, 2]
[6827, 1]
[6829, 2]
[6833, 1]
[6841, 2]
[6857, 0]
[6863, 0]
[6869, 1]
[6871, 0]
[6883, 2]
[6899, 1]
[6907, 2]
[6911, 0]
[6917, 0]
[6947, 1]
[6949, 2]
[6959, 1]
[6961, 0]
[6967, 2]
[6971, 0]
[6977, 0]
[6983, 1]
[6991, 0]
[6997, 2]
[7001, 0]
[7013, 0]
[7019, 1]
[7027, 0]
[7039, 2]
[7043, 1]
[7057, 0]
[7069, 2]
[7079, 0]
[7103, 0]
[7109, 0]
[7121, 0]
[7127, 1]
[7129, 2]
[7151, 1]
[7159, 0]
[7177, 2]
[7187, 0]
[7193, 1]
[7207, 2]
[7211, 1]
[7213, 0]
[7219, 2]
[7229, 1]
[7237, 0]
[7243, 2]
[7247, 0]
[7253, 0]
[7283, 1]
[7297, 2]
[7307, 1]
[7309, 0]
[7321, 2]
[7331, 1]
[7333, 2]
[7349, 1]
[7351, 0]
[7369, 0]
[7393, 0]
[7411, 0]
[7417, 2]
[7433, 0]
[7451, 0]
[7457, 1]
[7459, 0]
[7477, 2]
[7481, 0]
[7487, 1]
[7489, 2]
[7499, 1]
[7507, 2]
[7517, 0]
[7523, 0]
[7529, 1]
[7537, 2]
[7541, 0]
[7547, 1]
[7549, 2]
[7559, 1]
[7561, 0]
[7573, 2]
[7577, 0]
[7583, 0]
[7589, 1]
[7591, 0]
[7603, 2]
[7607, 1]
[7621, 0]
[7639, 2]
[7643, 0]
[7649, 1]
[7669, 2]
[7673, 1]
[7681, 0]
[7687, 2]
[7691, 1]
[7699, 2]
[7703, 1]
[7717, 0]
[7723, 2]
[7727, 1]
[7741, 0]
[7753, 2]
[7757, 1]
[7759, 0]
[7789, 2]
[7793, 0]
[7817, 0]
[7823, 0]
[7829, 0]
[7841, 0]
[7853, 1]
[7867, 0]
[7873, 2]
[7877, 1]
[7879, 2]
[7883, 0]
[7901, 0]
[7907, 0]
[7919, 1]
[7927, 0]
[7933, 2]
[7937, 0]
[7949, 1]
[7951, 0]
[7963, 0]
[7993, 2]
[8009, 1]
[8011, 0]
[8017, 2]
[8039, 1]
[8053, 0]
[8059, 2]
[8069, 0]
[8081, 0]
[8087, 1]
[8089, 2]
[8093, 1]
[8101, 2]
[8111, 0]
[8117, 0]
[8123, 0]
[8147, 1]
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With the first condition's initial part, you get
$$\begin{equation}\begin{aligned} \frac{p_2-p_1}{2} & \equiv 1 \pmod 3 \\ p_2 - p_1 & \equiv 2 \pmod 3 \end{aligned}\end{equation}\tag{1}\label{eq1A}$$
Since $p_2$ and $p_1$ are primes greater than $3$, if $p_2 \equiv 2 \pmod 3$, then $p_1 \equiv 0 \pmod 3$, which is not allowed. Thus, $p_2 \equiv 1 \pmod 3$ so then $p_1 \equiv 2 \pmod 3$.
Similarly, consider that you have
$$\begin{equation}\begin{aligned} \frac{p_3-p_2}{2} & \equiv 1 \pmod 3 \\ p_3 - p_2 & \equiv 2 \pmod 3 \end{aligned}\end{equation}\tag{2}\label{eq2A}$$
As before, this gives $p_3 \equiv 1 \pmod 3$ and $p_2 \equiv 2 \pmod 3$. However, you can't have $p_2 \equiv 1 \pmod 3$ from before and $p_2 \equiv 2 \pmod 3$ here. This is why your first set of conditions always hold.
You can use basically the same reasoning to show your second set of conditions always hold as well. I'll leave that to you to do yourself.
Note this answer only uses that none of $p_1$, $p_2$ and $p_3$ are a multiple of $3$. Thus, your $2$ sets of conditions apply to any $3$ primes in any order, as well as to non-primes.