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Theorem rexlimdvv 2439
 Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
Hypothesis
Ref Expression
rexlimdvv.1 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
Assertion
Ref Expression
rexlimdvv (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝑦,𝜑   𝜒,𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rexlimdvv
StepHypRef Expression
1 rexlimdvv.1 . . . 4 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
21expdimp 246 . . 3 ((𝜑𝑥𝐴) → (𝑦𝐵 → (𝜓𝜒)))
32rexlimdv 2432 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓𝜒))
43rexlimdva 2433 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  rexlimdvva  2440  f1oiso2  5466  xpdom2  6305  genpcdl  6617  genpcuu  6618  distrlem1prl  6680  distrlem1pru  6681  distrlem5prl  6684  distrlem5pru  6685  recexprlemss1l  6733  recexprlemss1u  6734  qaddcl  8570  qmulcl  8572
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