Theorem rexlimiva 2643

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 2642	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   E.wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  unon  4603  reg2exmidlema  4626  ssfilem  7045  diffitest  7057  fival  7148  elfi2  7150  fi0  7153  djuss  7248  updjud  7260  enumct  7293  finnum  7366  dmaddpqlem  7575  nqpi  7576  nq0nn  7640  recexprlemm  7822  iswrd  11086  wrdf  11090  rexanuz  11515  r19.2uz  11520  maxleast  11740  fsum2dlemstep  11961  fisumcom2  11965  fprod2dlemstep  12149  fprodcom2fi  12153  0dvds  12338  even2n  12401  m1expe  12426  m1exp1  12428  modprm0  12793  gsumval2  13446  dfgrp2  13576  epttop  14780  neipsm  14844  tgioo  15244  sin0pilem2  15472  pilem3  15473  perfect  15691  clwwlkn1loopb  16162  bj-nn0suc  16410  bj-nn0sucALT  16424  trirec0xor  16501