Theorem pimeqed ≪ | index | src | ≫ 

theorem pimeqed (G: wff) {x: nat} (a: nat) (p: wff x) (q: wff):
  $ G /\ x = a -> (p <-> q) $ >
  $ G -> ((P. x x = a -> p) <-> q) $;
StepHypRefExpression
1 bian1 
E. x x = a -> (E. x x = a /\ A. x (x = a -> p) <-> A. x (x = a -> p))
2 1 conv pim 
E. x x = a -> ((P. x x = a -> p) <-> A. x (x = a -> p))
3 ax_6 
E. x x = a
4 2, 3 ax_mp 
(P. x x = a -> p) <-> A. x (x = a -> p)
5 hyp e 
G /\ x = a -> (p <-> q)
6 5 aleqed 
G -> (A. x (x = a -> p) <-> q)
7 4, 6 syl5bb 
G -> ((P. x x = a -> p) <-> q)

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12)