Theorem rexbida 2321

Description: Formula-building rule for restricted existential quantifier (deduction rule). (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 425	. . 3 |- ( ph -> ( ( x e. A /\ ps ) <-> ( x e. A /\ ch ) ) )
4	1, 3	exbid 1507	. 2 |- ( ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ ch ) ) )
5		df-rex 2312	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
6		df-rex 2312	. 2 |- ( E. x e. A ch <-> E. x ( x e. A /\ ch ) )
7	4, 5, 6	3bitr4g 212	1 |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   F/wnf 1349   E.wex 1381   e. wcel 1393   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-rex 2312

This theorem is referenced by:  rexbidva  2323  rexbid  2325  rexbi  2446  dfiun2g  3689  fun11iun  5147