Theorem bian1rexa ≪ | index | src | ≫ 

theorem bian1rexa (b: wff) {x: nat} (p a c: wff x):
  $ p -> (a <-> b /\ c) $ >
  $ E. x (p /\ a) <-> b /\ E. x (p /\ c) $;
StepHypRefExpression
1 bitr 
(E. x (p /\ a) <-> E. x (p /\ (b /\ c))) -> (E. x (p /\ (b /\ c)) <-> b /\ E. x (p /\ c)) -> (E. x (p /\ a) <-> b /\ E. x (p /\ c))
2 hyp h 
p -> (a <-> b /\ c)
3 2 rexeqa 
E. x (p /\ a) <-> E. x (p /\ (b /\ c))
4 1, 3 ax_mp 
(E. x (p /\ (b /\ c)) <-> b /\ E. x (p /\ c)) -> (E. x (p /\ a) <-> b /\ E. x (p /\ c))
5 rexan1 
E. x (p /\ (b /\ c)) <-> b /\ E. x (p /\ c)
6 4, 5 ax_mp 
E. x (p /\ a) <-> b /\ E. x (p /\ c)

Axiom use

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