Theorem rexlimiva 2663

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 2662	1 |- ( E. x e. A ph -> ps )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   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-17 1579  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:  unon  4658  reg2exmidlema  4681  ssfilem  7177  ssfilemd  7179  diffitest  7191  fival  7304  elfi2  7306  fi0  7309  djuss  7410  updjud  7422  enumct  7455  finnum  7528  dmaddpqlem  7744  nqpi  7745  nq0nn  7809  recexprlemm  7991  iswrd  11306  wrdf  11310  rexanuz  11754  r19.2uz  11759  maxleast  11979  fsum2dlemstep  12201  fisumcom2  12205  fprod2dlemstep  12389  fprodcom2fi  12393  0dvds  12578  even2n  12641  m1expe  12666  m1exp1  12668  modprm0  13033  gzsumval2  13714  dfgrp2  13832  epttop  15191  neipsm  15255  tgioo  15655  sin0pilem2  15883  pilem3  15884  perfect  16115  clwwlkn1loopb  16661  bj-nn0suc  16990  bj-nn0sucALT  17004  trirec0xor  17094