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Theorem rexlimivv 2587
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
Hypothesis
Ref Expression
rexlimivv.1 ((𝑥𝐴𝑦𝐵) → (𝜑𝜓))
Assertion
Ref Expression
rexlimivv (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
Distinct variable groups:   𝑥,𝑦,𝜓   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rexlimivv
StepHypRef Expression
1 rexlimivv.1 . . 3 ((𝑥𝐴𝑦𝐵) → (𝜑𝜓))
21rexlimdva 2581 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑𝜓))
32rexlimiv 2575 1 (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2135  wrex 2443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-ral 2447  df-rex 2448
This theorem is referenced by:  opelxp  4631  f1o2ndf1  6190  xpdom2  6791  distrlem5prl  7521  distrlem5pru  7522  mulid1  7890  cnegex  8070  recexap  8544  creur  8848  creui  8849  cju  8850  elz2  9256  qre  9557  qaddcl  9567  qnegcl  9568  qmulcl  9569  qreccl  9574  elpqb  9581  replim  10795  prodmodc  11513  odd2np1  11804  opoe  11826  omoe  11827  opeo  11828  omeo  11829  qredeu  12023  pythagtriplem1  12191  pcz  12257  4sqlem1  12312  4sqlem2  12313  4sqlem4  12316  mul4sq  12318  txuni2  12854  blssioo  13143  tgioo  13144
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