Theorem birexan2a ≪ | index | src | ≫ 

theorem birexan2a (a: wff) {x: nat} (p b c: wff x):
  $ a -> (b <-> E. x (p /\ c)) $ >
  $ a /\ b <-> E. x (p /\ (a /\ c)) $;
StepHypRefExpression
1 bitr 
(a /\ b <-> E. x (a /\ (p /\ c))) -> (E. x (a /\ (p /\ c)) <-> E. x (p /\ (a /\ c))) -> (a /\ b <-> E. x (p /\ (a /\ c)))
2 hyp h 
a -> (b <-> E. x (p /\ c))
3 2 biexan2a 
a /\ b <-> E. x (a /\ (p /\ c))
4 1, 3 ax_mp 
(E. x (a /\ (p /\ c)) <-> E. x (p /\ (a /\ c))) -> (a /\ b <-> E. x (p /\ (a /\ c)))
5 anlass 
a /\ (p /\ c) <-> p /\ (a /\ c)
6 5 exeqi 
E. x (a /\ (p /\ c)) <-> E. x (p /\ (a /\ c))
7 4, 6 ax_mp 
a /\ b <-> E. x (p /\ (a /\ c))

Axiom use

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