Theorem rexlimiva 2544

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 2543	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   E.wrex 2417

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 2421  df-rex 2422

This theorem is referenced by:  unon  4427  reg2exmidlema  4449  ssfilem  6769  diffitest  6781  fival  6858  elfi2  6860  fi0  6863  djuss  6955  updjud  6967  enumct  7000  finnum  7039  dmaddpqlem  7185  nqpi  7186  nq0nn  7250  recexprlemm  7432  rexanuz  10760  r19.2uz  10765  maxleast  10985  fsum2dlemstep  11203  fisumcom2  11207  0dvds  11513  even2n  11571  m1expe  11596  m1exp1  11598  epttop  12259  neipsm  12323  tgioo  12715  sin0pilem2  12863  pilem3  12864  bj-nn0suc  13162  bj-nn0sucALT  13176