Theorem rexbid2 2475

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)

Hypotheses

Ref	Expression
rexbid2.nf	|- F/ x ph
rexbid2.1	|- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )

Assertion

Ref	Expression
rexbid2	|- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Proof of Theorem rexbid2

Step	Hyp	Ref	Expression
1		rexbid2.nf	. . 3 |- F/ x ph
2		rexbid2.1	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. B /\ ch ) ) )
3	1, 2	exbid 1609	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. B /\ ch ) ) )
4		df-rex 2454	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
5		df-rex 2454	. 2 |- ( E. x e. B ch <-> E. x ( x e. B /\ ch ) )
6	3, 4, 5	3bitr4g 222	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   F/wnf 1453   E.wex 1485   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-rex 2454

This theorem is referenced by: (None)