Theorem rexlimiv 2642

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Hypothesis

Ref	Expression
rexlimiv.1	|- ( x e. A -> ( ph -> ps ) )

Assertion

Ref	Expression
rexlimiv	|- ( E. x e. A ph -> ps )

Distinct variable group:   ps, x

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem rexlimiv

Step	Hyp	Ref	Expression
1		nfv 1574	. 2 |- F/ x ps
2		rexlimiv.1	. 2 |- ( x e. A -> ( ph -> ps ) )
3	1, 2	rexlimi 2641	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   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:  rexlimiva  2643  rexlimivw  2644  rexlimivv  2654  r19.36av  2682  r19.44av  2690  r19.45av  2691  rexn0  3590  uniss2  3919  elres  5041  ssimaex  5695  mpoexw  6359  tfrlem5  6460  tfrlem8  6464  ecoptocl  6769  mapsn  6837  elixpsn  6882  ixpsnf1o  6883  findcard  7050  findcard2  7051  findcard2s  7052  fiintim  7093  prnmaddl  7677  0re  8146  cnegexlem2  8322  0cnALT  8336  bndndx  9368  uzn0  9738  ublbneg  9808  rexanuz2  11502  opnneiid  14838  2lgslem1b  15768  2sqlem2  15794  bj-inf2vnlem2  16334