Theorem rexlimiva 2657

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)

Hypothesis

Ref	Expression
rexlimiva.1	|- ( ( x e. A /\ ph ) -> ps )

Assertion

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

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rexlimiva

Step	Hyp	Ref	Expression
1		rexlimiva.1	. . 3 |- ( ( x e. A /\ ph ) -> ps )
2	1	ex 115	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	rexlimiv 2656	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   E.wrex 2523

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2527  df-rex 2528

This theorem is referenced by:  unon  4638  reg2exmidlema  4661  ssfilem  7143  ssfilemd  7145  diffitest  7157  fival  7270  elfi2  7272  fi0  7275  djuss  7374  updjud  7386  enumct  7419  finnum  7492  dmaddpqlem  7708  nqpi  7709  nq0nn  7773  recexprlemm  7955  iswrd  11251  wrdf  11255  rexanuz  11698  r19.2uz  11703  maxleast  11923  fsum2dlemstep  12145  fisumcom2  12149  fprod2dlemstep  12333  fprodcom2fi  12337  0dvds  12522  even2n  12585  m1expe  12610  m1exp1  12612  modprm0  12977  gsumval2  13660  dfgrp2  13782  epttop  15081  neipsm  15145  tgioo  15545  sin0pilem2  15773  pilem3  15774  perfect  15995  clwwlkn1loopb  16541  bj-nn0suc  16860  bj-nn0sucALT  16874  trirec0xor  16955