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Theorem rexlimivv 2555
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 2549 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑𝜓))
32rexlimiv 2543 1 (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1480  wrex 2417
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2421  df-rex 2422
This theorem is referenced by:  opelxp  4569  f1o2ndf1  6125  xpdom2  6725  distrlem5prl  7401  distrlem5pru  7402  mulid1  7770  cnegex  7947  recexap  8421  creur  8724  creui  8725  cju  8726  elz2  9129  qre  9424  qaddcl  9434  qnegcl  9435  qmulcl  9436  qreccl  9441  replim  10638  prodmodc  11354  odd2np1  11577  opoe  11599  omoe  11600  opeo  11601  omeo  11602  qredeu  11785  txuni2  12435  blssioo  12724  tgioo  12725
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