Theorem rexlimi 2470

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 2416	. 2 |- A. x e. A ( ph -> ps )
3		rexlimi.1	. . 3 |- F/ x ps
4	3	r19.23 2468	. 2 |- ( A. x e. A ( ph -> ps ) <-> ( E. x e. A ph -> ps ) )
5	2, 4	mpbi 143	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   F/wnf 1389   e. wcel 1433   A.wral 2348   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  rexlimiv  2471  r19.29af2  2496  triun  3888  reusv1  4208  reusv3  4210  onintrab2im  4262  fun11iun  5167