Theorem sbseq2d | index | src |

theorem sbseq2d {x: nat} (G: wff) (a: nat x) (A B: set x):
  $ G -> A == B $ >
  $ G -> S[a / x] A == S[a / x] B $;
StepHypRefExpression
1 hyp h
G -> A == B
2 1 eleq2d
G -> (y e. A <-> y e. B)
3 2 sbeq2d
G -> ([a / x] y e. A <-> [a / x] y e. B)
4 3 abeqd
G -> {y | [a / x] y e. A} == {y | [a / x] y e. B}
5 4 conv sbs
G -> S[a / x] A == S[a / x] 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)