Theorem pimeq1 ≪ | index | src | ≫ 

theorem pimeq1 {x: nat} (p1 p2 q: wff x):
  $ A. x (p1 <-> p2) -> ((P. x p1 -> q) <-> (P. x p2 -> q)) $;
StepHypRefExpression
1 exeq 
A. x (p1 <-> p2) -> (E. x p1 <-> E. x p2)
2 aleq 
A. x (p1 -> q <-> p2 -> q) -> (A. x (p1 -> q) <-> A. x (p2 -> q))
3 id 
(p1 <-> p2) -> (p1 <-> p2)
4 3 imeq1d 
(p1 <-> p2) -> (p1 -> q <-> p2 -> q)
5 4 alimi 
A. x (p1 <-> p2) -> A. x (p1 -> q <-> p2 -> q)
6 2, 5 syl 
A. x (p1 <-> p2) -> (A. x (p1 -> q) <-> A. x (p2 -> q))
7 1, 6 aneqd 
A. x (p1 <-> p2) -> (E. x p1 /\ A. x (p1 -> q) <-> E. x p2 /\ A. x (p2 -> q))
8 7 conv pim 
A. x (p1 <-> p2) -> ((P. x p1 -> q) <-> (P. x p2 -> q))

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4)