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Database ZF (ZERMELO-FRAENKEL) SET THEORY ZF Set Theory - start with the Axiom of Extensionality Restricted quantification Restricted universal and existential quantification raleq
Description: Equality theorem for restricted universal quantifier. (Contributed by NM , 16-Nov-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
raleq |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
Proof
Step
Hyp
Ref
Expression
1
rexeq |- ( A = B -> ( E. x e. A -. ph <-> E. x e. B -. ph ) )
2
rexnal |- ( E. x e. A -. ph <-> -. A. x e. A ph )
3
rexnal |- ( E. x e. B -. ph <-> -. A. x e. B ph )
4
1 2 3
3bitr3g |- ( A = B -> ( -. A. x e. A ph <-> -. A. x e. B ph ) )
5
4
con4bid |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )