Theorem rexlimi 2661

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 2603	. 2 |- A. x e. A ( ph -> ps )
3		rexlimi.1	. . 3 |- F/ x ps
4	3	r19.23 2659	. 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 )

This proof depends on syntax axioms:   -> wi 4   F/wnf 1513   e. wcel 2209   A.wral 2528   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  rexlimiv  2662  r19.29af2  2691  triun  4242  reusv1  4604  reusv3  4606  onintrab2im  4665  fun11iun  5660  fisumcom2  12205  fprodcom2fi  12393