Theorem rexlimiv 2588

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 1528	. 2 |- F/ x ps
2		rexlimiv.1	. 2 |- ( x e. A -> ( ph -> ps ) )
3	1, 2	rexlimi 2587	1 |- ( E. x e. A ph -> ps )

Syntax hints:   -> wi 4   e. wcel 2148   E.wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  rexlimiva  2589  rexlimivw  2590  rexlimivv  2600  r19.36av  2628  r19.44av  2636  r19.45av  2637  rexn0  3521  uniss2  3840  elres  4943  ssimaex  5577  mpoexw  6213  tfrlem5  6314  tfrlem8  6318  ecoptocl  6621  mapsn  6689  elixpsn  6734  ixpsnf1o  6735  findcard  6887  findcard2  6888  findcard2s  6889  fiintim  6927  prnmaddl  7488  0re  7956  cnegexlem2  8132  0cnALT  8146  bndndx  9174  uzn0  9542  ublbneg  9612  rexanuz2  10999  opnneiid  13634  2sqlem2  14432  bj-inf2vnlem2  14693