Theorem exifp ≪ | index | src | ≫ 

theorem exifp (p: wff) {x: nat} (a b: wff x):
  $ E. x ifp p a b <-> ifp p (E. x a) (E. x b) $;
StepHypRefExpression
1 bitr 
(E. x ifp p a b <-> E. x (p /\ a) \/ E. x (~p /\ b)) ->
  (E. x (p /\ a) \/ E. x (~p /\ b) <-> ifp p (E. x a) (E. x b)) ->
  (E. x ifp p a b <-> ifp p (E. x a) (E. x b))
2 exor 
E. x (p /\ a \/ ~p /\ b) <-> E. x (p /\ a) \/ E. x (~p /\ b)
3 2 conv ifp 
E. x ifp p a b <-> E. x (p /\ a) \/ E. x (~p /\ b)
4 1, 3 ax_mp 
(E. x (p /\ a) \/ E. x (~p /\ b) <-> ifp p (E. x a) (E. x b)) -> (E. x ifp p a b <-> ifp p (E. x a) (E. x b))
5 oreq 
(E. x (p /\ a) <-> p /\ E. x a) -> (E. x (~p /\ b) <-> ~p /\ E. x b) -> (E. x (p /\ a) \/ E. x (~p /\ b) <-> p /\ E. x a \/ ~p /\ E. x b)
6 5 conv ifp 
(E. x (p /\ a) <-> p /\ E. x a) -> (E. x (~p /\ b) <-> ~p /\ E. x b) -> (E. x (p /\ a) \/ E. x (~p /\ b) <-> ifp p (E. x a) (E. x b))
7 exan1 
E. x (p /\ a) <-> p /\ E. x a
8 6, 7 ax_mp 
(E. x (~p /\ b) <-> ~p /\ E. x b) -> (E. x (p /\ a) \/ E. x (~p /\ b) <-> ifp p (E. x a) (E. x b))
9 exan1 
E. x (~p /\ b) <-> ~p /\ E. x b
10 8, 9 ax_mp 
E. x (p /\ a) \/ E. x (~p /\ b) <-> ifp p (E. x a) (E. x b)
11 4, 10 ax_mp 
E. x ifp p a b <-> ifp p (E. x a) (E. x b)

Axiom use

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