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  4660  reg2exmidlema  4683  ssfilem  7183  ssfilemd  7185  diffitest  7197  fival  7309  elfi2  7311  fi0  7314  djuss  7416  updjud  7428  enumct  7461  finnum  7534  dmaddpqlem  7750  nqpi  7751  nq0nn  7815  recexprlemm  7997  iswrd  11327  wrdf  11331  rexanuz  11775  r19.2uz  11780  maxleast  12001  fsum2dlemstep  12225  fisumcom2  12229  fprod2dlemstep  12413  fprodcom2fi  12417  0dvds  12602  even2n  12665  m1expe  12690  m1exp1  12692  modprm0  13061  gzsumval2  13772  dfgrp2  13890  epttop  15333  neipsm  15397  tgioo  15797  sin0pilem2  16026  pilem3  16027  perfect  16318  clwwlkn1loopb  16883  bj-nn0suc  17212  bj-nn0sucALT  17226  trirec0xor  17318