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Theorem rexlimivv 2532
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 2526 . 2 (𝑥𝐴 → (∃𝑦𝐵 𝜑𝜓))
32rexlimiv 2520 1 (∃𝑥𝐴𝑦𝐵 𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1465  wrex 2394
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-ral 2398  df-rex 2399
This theorem is referenced by:  opelxp  4539  f1o2ndf1  6093  xpdom2  6693  distrlem5prl  7362  distrlem5pru  7363  mulid1  7731  cnegex  7908  recexap  8382  creur  8685  creui  8686  cju  8687  elz2  9090  qre  9385  qaddcl  9395  qnegcl  9396  qmulcl  9397  qreccl  9402  replim  10599  odd2np1  11497  opoe  11519  omoe  11520  opeo  11521  omeo  11522  qredeu  11705  txuni2  12352  blssioo  12641  tgioo  12642
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