Theorem rexlimiva 2577

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 114	. 2 |- ( x e. A -> ( ph -> ps ) )
3	2	rexlimiv 2576	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   E.wrex 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2448  df-rex 2449

This theorem is referenced by:  unon  4487  reg2exmidlema  4510  ssfilem  6837  diffitest  6849  fival  6931  elfi2  6933  fi0  6936  djuss  7031  updjud  7043  enumct  7076  finnum  7135  dmaddpqlem  7314  nqpi  7315  nq0nn  7379  recexprlemm  7561  rexanuz  10926  r19.2uz  10931  maxleast  11151  fsum2dlemstep  11371  fisumcom2  11375  fprod2dlemstep  11559  fprodcom2fi  11563  0dvds  11747  even2n  11807  m1expe  11832  m1exp1  11834  modprm0  12182  epttop  12690  neipsm  12754  tgioo  13146  sin0pilem2  13303  pilem3  13304  bj-nn0suc  13806  bj-nn0sucALT  13820  trirec0xor  13884