Demonstrate that p ↔ (p ↔ q) ⇔ q

Question

I know the answer is :

(p ↔ p) ↔ q ⇔ q
1 ↔ q ⇔ q
q ⇔ q

But I don't understand why it isn't :

(p ↔ p) ↔ q ⇔ q
1 ↔ q ⇔ q
(1 → q) ^ (q → 1) ⇔ q
(1 → q) ^ 1 ⇔ q
1 ⇔ q // fail !

Thanks.

Answer 1

Your mistake is here: (1 → q) ^ 1 should become (1 → q), not 1. So, we'd be left with

       (1 → q) ⇔ q

Which leads us to the desired conclusion, q ⇔ q.