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Metamath Proof Explorer

Theorem rexeq

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Shorten other proofs. (Revised by Wolf Lammen, 8-Mar-2025)

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

Proof

Step Hyp Ref Expression
1 dfcleq 
 |-  ( A = B <-> A. x ( x e. A <-> x e. B ) )
2 anbi1 
 |-  ( ( x e. A <-> x e. B ) -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
3 2 alexbii 
 |-  ( A. x ( x e. A <-> x e. B ) -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
4 1 3 sylbi 
 |-  ( A = B -> ( E. x ( x e. A /\ ph ) <-> E. x ( x e. B /\ ph ) ) )
5 df-rex 
 |-  ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6 df-rex 
 |-  ( E. x e. B ph <-> E. x ( x e. B /\ ph ) )
7 4 5 6 3bitr4g 
 |-  ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )