Theorem rexbida 2465

Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
ralbida.1	|- F/ x ph
ralbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem rexbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 |- F/ x ph
2		ralbida.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	pm5.32da 449	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	exbid 1609	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ ch ) ) )
5		df-rex 2454	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
6		df-rex 2454	. 2 |- ( E. x e. A ch <-> E. x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 222	1 |- ( ph -> ( E. x e. A ps <-> E. x e. A 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:  rexbidva  2467  rexbid  2469  rexbi  2603  dfiun2g  3905  fun11iun  5463  ismkvnex  7131  mkvprop  7134