Theorem eleq2 | index | src |

theorem eleq2 (A B: set) (a: nat): $ A == B -> (a e. A <-> a e. B) $;
StepHypRefExpression
1 eleq1
x = a -> (x e. A <-> a e. A)
2 eleq1
x = a -> (x e. B <-> a e. B)
3 1, 2 bieqd
x = a -> (x e. A <-> x e. B <-> (a e. A <-> a e. B))
4 3 eale
A. x (x e. A <-> x e. B) -> (a e. A <-> a e. B)
5 4 conv eqs
A == B -> (a e. A <-> a e. B)

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_12), axs_set (ax_8)