Theorem cbvsbh ≪ | index | src | ≫ 

theorem cbvsbh {x y: nat} (a: nat) (p q: wff x y):
  $ F/ y p $ >
  $ F/ x q $ >
  $ x = y -> (p <-> q) $ >
  $ [a / x] p <-> [a / y] q $;
StepHypRefExpression
1 nfv 
F/ y x = z
2 hyp h1 
F/ y p
3 1, 2 nfim 
F/ y x = z -> p
4 nfv 
F/ x y = z
5 hyp h2 
F/ x q
6 4, 5 nfim 
F/ x y = z -> q
7 eqeq1 
x = y -> (x = z <-> y = z)
8 hyp e 
x = y -> (p <-> q)
9 7, 8 imeqd 
x = y -> (x = z -> p <-> y = z -> q)
10 3, 6, 9 cbvalh 
A. x (x = z -> p) <-> A. y (y = z -> q)
11 10 imeq2i 
z = a -> A. x (x = z -> p) <-> z = a -> A. y (y = z -> q)
12 11 aleqi 
A. z (z = a -> A. x (x = z -> p)) <-> A. z (z = a -> A. y (y = z -> q))
13 12 conv sb 
[a / x] p <-> [a / y] q

Axiom use

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