Theorem birexan1a ≪ | index | src | ≫ 

theorem birexan1a (c: wff) {x: nat} (p a b: wff x):
  $ c -> (a <-> E. x (p /\ b)) $ >
  $ a /\ c <-> E. x (p /\ (b /\ c)) $;
StepHypRefExpression
1 bitr 
(a /\ c <-> E. x (p /\ b /\ c)) -> (E. x (p /\ b /\ c) <-> E. x (p /\ (b /\ c))) -> (a /\ c <-> E. x (p /\ (b /\ c)))
2 hyp h 
c -> (a <-> E. x (p /\ b))
3 2 biexan1a 
a /\ c <-> E. x (p /\ b /\ c)
4 1, 3 ax_mp 
(E. x (p /\ b /\ c) <-> E. x (p /\ (b /\ c))) -> (a /\ c <-> E. x (p /\ (b /\ c)))
5 anass 
p /\ b /\ c <-> p /\ (b /\ c)
6 5 exeqi 
E. x (p /\ b /\ c) <-> E. x (p /\ (b /\ c))
7 4, 6 ax_mp 
a /\ c <-> E. x (p /\ (b /\ c))

Axiom use

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