Metamath Proof Explorer


Theorem rmoeq1OLD

Description: Obsolete version of rmoeq1 as of 12-Mar-2025. (Contributed by Alexander van der Vekens, 17-Jun-2017) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion rmoeq1OLD
|- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )

Proof

Step Hyp Ref Expression
1 eleq2
 |-  ( A = B -> ( x e. A <-> x e. B ) )
2 1 anbi1d
 |-  ( A = B -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
3 2 mobidv
 |-  ( A = B -> ( E* x ( x e. A /\ ph ) <-> E* x ( x e. B /\ ph ) ) )
4 df-rmo
 |-  ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
5 df-rmo
 |-  ( E* x e. B ph <-> E* x ( x e. B /\ ph ) )
6 3 4 5 3bitr4g
 |-  ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )