Theorem rexlimi 2597

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimi.1	|- F/ x ps
rexlimi.2	|- ( x e. A -> ( ph -> ps ) )

Assertion

Ref	Expression
rexlimi	|- ( E. x e. A ph -> ps )

Proof of Theorem rexlimi

Step	Hyp	Ref	Expression
1		rexlimi.2	. . 3 |- ( x e. A -> ( ph -> ps ) )
2	1	rgen 2540	. 2 |- A. x e. A ( ph -> ps )
3		rexlimi.1	. . 3 |- F/ x ps
4	3	r19.23 2595	. 2 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )
5	2, 4	mpbi 145	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   F/wnf 1470   e. wcel 2158   A.wral 2465   E.wrex 2466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-ral 2470  df-rex 2471

This theorem is referenced by:  rexlimiv  2598  r19.29af2  2627  triun  4126  reusv1  4470  reusv3  4472  onintrab2im  4529  fun11iun  5494  fisumcom2  11460  fprodcom2fi  11648