Theorem elsab ≪ | index | src | ≫ 

theorem elsab (B: set) (a b: nat) {x: nat} (A: set x):
  $ x = a -> A == B $ >
  $ a, b e. S\ x, A <-> b e. B $;
StepHypRefExpression
1 bitr 
(a, b e. S\ x, A <-> b e. S[a / x] A) -> (b e. S[a / x] A <-> b e. B) -> (a, b e. S\ x, A <-> b e. B)
2 elsabs 
a, b e. S\ x, A <-> b e. S[a / x] A
3 1, 2 ax_mp 
(b e. S[a / x] A <-> b e. B) -> (a, b e. S\ x, A <-> b e. B)
4 eleq2 
S[a / x] A == B -> (b e. S[a / x] A <-> b e. B)
5 hyp h 
x = a -> A == B
6 5 sbse 
S[a / x] A == B
7 4, 6 ax_mp 
b e. S[a / x] A <-> b e. B
8 3, 7 ax_mp 
a, b e. S\ x, A <-> b e. 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), axs_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)