Theorem rexlimiva 2655

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 2654	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   E.wrex 2521

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 2525  df-rex 2526

This theorem is referenced by:  unon  4633  reg2exmidlema  4656  ssfilem  7130  ssfilemd  7132  diffitest  7144  fival  7257  elfi2  7259  fi0  7262  djuss  7361  updjud  7373  enumct  7406  finnum  7479  dmaddpqlem  7692  nqpi  7693  nq0nn  7757  recexprlemm  7939  iswrd  11226  wrdf  11230  rexanuz  11673  r19.2uz  11678  maxleast  11898  fsum2dlemstep  12120  fisumcom2  12124  fprod2dlemstep  12308  fprodcom2fi  12312  0dvds  12497  even2n  12560  m1expe  12585  m1exp1  12587  modprm0  12952  gsumval2  13610  dfgrp2  13740  epttop  14955  neipsm  15019  tgioo  15419  sin0pilem2  15647  pilem3  15648  perfect  15869  clwwlkn1loopb  16415  bj-nn0suc  16734  bj-nn0sucALT  16748  trirec0xor  16829