Theorem rexbid 3051

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)

Hypotheses

Ref	Expression
rexbid.1	|- F/ x ph
rexbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem rexbid

Step	Hyp	Ref	Expression
1		rexbid.1	. 2 |- F/ x ph
2		rexbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	2	adantr 481	. 2 |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
4	1, 3	rexbida 3047	1 |- ( ph -> ( E. x e. A ps <-> E. x e. A ch ) )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   e. wcel 1990   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  rexbidvALT  3053  rexeqbid  3151  scott0  8749  infcvgaux1i  14589  bnj1463  31123  poimirlem25  33434  poimirlem26  33435  elrnmptf  39367  smfsupmpt  41021  smfinfmpt  41025