Theorem biexan1a ≪ | index | src | ≫ 

theorem biexan1a (c: wff) {x: nat} (a b: wff x):
  $ c -> (a <-> E. x b) $ >
  $ a /\ c <-> E. x (b /\ c) $;
StepHypRefExpression
1 bitr4 
(a /\ c <-> E. x b /\ c) -> (E. x (b /\ c) <-> E. x b /\ c) -> (a /\ c <-> E. x (b /\ c))
2 aneq1a 
(c -> (a <-> E. x b)) -> (a /\ c <-> E. x b /\ c)
3 hyp h 
c -> (a <-> E. x b)
4 2, 3 ax_mp 
a /\ c <-> E. x b /\ c
5 1, 4 ax_mp 
(E. x (b /\ c) <-> E. x b /\ c) -> (a /\ c <-> E. x (b /\ c))
6 exan2 
E. x (b /\ c) <-> E. x b /\ c
7 5, 6 ax_mp 
a /\ c <-> E. x (b /\ c)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp), axs_pred_calc (ax_gen, ax_4, ax_5)