How can you prove that a function in z is Holomorphic(infinitely differentiable along all paths) through Cauchy Reimann conditions(proof: x, y only)

Question

According to what I read, any function satisfying the cauchy reimann conditions is holomorphic in nature whose proof comes by expanding Delta(z) in terms of Delta(x) and Delta(y) and considering only 2 paths along x and y axes or along real and imaginary axes for which the limits are set to approach 0. Now if we approach the z = 0 from some random path not along the axes, how can you prove that the function is still holomorphic using the the cauchy reimann conditions, or should it even suffice as the proof was valid only from the limit along the axes?