Theorem biexan2a ≪ | index | src | ≫ 

theorem biexan2a (a: wff) {x: nat} (b c: wff x):
  $ a -> (b <-> E. x c) $ >
  $ a /\ b <-> E. x (a /\ c) $;
StepHypRefExpression
1 bitr4 
(a /\ b <-> a /\ E. x c) -> (E. x (a /\ c) <-> a /\ E. x c) -> (a /\ b <-> E. x (a /\ c))
2 aneq2a 
(a -> (b <-> E. x c)) -> (a /\ b <-> a /\ E. x c)
3 hyp h 
a -> (b <-> E. x c)
4 2, 3 ax_mp 
a /\ b <-> a /\ E. x c
5 1, 4 ax_mp 
(E. x (a /\ c) <-> a /\ E. x c) -> (a /\ b <-> E. x (a /\ c))
6 exan1 
E. x (a /\ c) <-> a /\ E. x c
7 5, 6 ax_mp 
a /\ b <-> E. x (a /\ c)

Axiom use

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