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  3523  uniss2  3842  elres  4945  ssimaex  5579  mpoexw  6216  tfrlem5  6317  tfrlem8  6321  ecoptocl  6624  mapsn  6692  elixpsn  6737  ixpsnf1o  6738  findcard  6890  findcard2  6891  findcard2s  6892  fiintim  6930  prnmaddl  7491  0re  7959  cnegexlem2  8135  0cnALT  8149  bndndx  9177  uzn0  9545  ublbneg  9615  rexanuz2  11002  opnneiid  13749  2sqlem2  14547  bj-inf2vnlem2  14808