Theorem rexlimiv 2662

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 1581	. 2 |- F/ x ps
2		rexlimiv.1	. 2 |- ( x e. A -> ( ph -> ps ) )
3	1, 2	rexlimi 2661	1 |- ( E. x e. A ph -> ps )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  rexlimiva  2663  rexlimivw  2664  rexlimivv  2674  r19.36av  2702  r19.44av  2710  r19.45av  2711  rexn0  3626  uniss2  3966  elres  5101  ssimaex  5767  mpoexw  6453  tfrlem5  6590  tfrlem8  6594  ecoptocl  6901  mapsn  6978  elixpsn  7023  ixpsnf1o  7024  findcard  7198  findcard2  7199  findcard2s  7200  fiintim  7244  prnmaddl  7863  0re  8332  cnegexlem2  8509  0cnALT  8523  bndndx  9573  uzn0  9953  ublbneg  10028  rexanuz2  11778  opnneiid  15407  2lgslem1b  16430  2sqlem2  16456  bj-inf2vnlem2  17219