Theorem cbvsabd ≪ | index | src | ≫ 

theorem cbvsabd {x y: nat} (G: wff) (A: set x) (B: set y):
  $ G /\ x = y -> A == B $ >
  $ G -> S\ x, A == S\ y, B $;
StepHypRefExpression
1 cbvsabs 
S\ x, A == S\ y, (S[y / x] A)
2 sbset 
A. x (x = y -> A == B) -> S[y / x] A == B
3 hyp h 
G /\ x = y -> A == B
4 3 ialda 
G -> A. x (x = y -> A == B)
5 2, 4 syl 
G -> S[y / x] A == B
6 5 sabeqd 
G -> S\ y, (S[y / x] A) == S\ y, B
7 1, 6 syl5eqs 
G -> S\ x, A == S\ y, B

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), axs_set (elab, ax_8)