Theorem rexlimi 2471

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 2417	. 2 |- A. x e. A ( ph -> ps )
3		rexlimi.1	. . 3 |- F/ x ps
4	3	r19.23 2469	. 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 1390   e. wcel 1434   A.wral 2349   E.wrex 2350

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354  df-rex 2355

This theorem is referenced by:  rexlimiv  2472  r19.29af2  2497  triun  3896  reusv1  4216  reusv3  4218  onintrab2im  4270  fun11iun  5178