Prove $A \equiv B \vdash (\forall x)A \equiv (\forall x)B$ without relying on equivalence theorem.

Question

The given rules of inference for me to use are here: rules of inference

In addition, I was taught the deduction theorem, the ping-pong tautology, specialization and dual specialization (derived rules).

My solution:

Using Deduction Theorem, it suffices if we prove $$A \equiv B, (\forall x)A \vdash (\forall x)B.$$ Hilbert proof:

1) $A ≡ B\quad\langle\text{hypothesis}\rangle$

2) $(\forall x)A\quad\langle\text{hypothesis}\rangle$

3) $A\quad\langle\text{2 + Specialization}\rangle$

I couldn't move past step 3 because I did not know where to go. Any help on this question is appreciated.

Note: This is practice problem is for first year discrete mathematics for engineers course.

Answer 1

I did this, not sure if it's correct or not.