Theorem rexlimiva 2542

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 2541	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   E.wrex 2415

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420

This theorem is referenced by:  unon  4422  reg2exmidlema  4444  ssfilem  6762  diffitest  6774  fival  6851  elfi2  6853  fi0  6856  djuss  6948  updjud  6960  enumct  6993  finnum  7032  dmaddpqlem  7178  nqpi  7179  nq0nn  7243  recexprlemm  7425  rexanuz  10753  r19.2uz  10758  maxleast  10978  fsum2dlemstep  11196  fisumcom2  11200  0dvds  11502  even2n  11560  m1expe  11585  m1exp1  11587  epttop  12248  neipsm  12312  tgioo  12704  sin0pilem2  12852  pilem3  12853  bj-nn0suc  13151  bj-nn0sucALT  13165