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Theorem rexlimdvv 2556
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 257 . . 3 ((𝜑𝑥𝐴) → (𝑦𝐵 → (𝜓𝜒)))
32rexlimdv 2548 . 2 ((𝜑𝑥𝐴) → (∃𝑦𝐵 𝜓𝜒))
43rexlimdva 2549 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
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:  rexlimdvva  2557  f1oiso2  5728  xpdom2  6725  genpcdl  7334  genpcuu  7335  distrlem1prl  7397  distrlem1pru  7398  distrlem5prl  7401  distrlem5pru  7402  recexprlemss1l  7450  recexprlemss1u  7451  qaddcl  9434  qmulcl  9436  summodc  11159  dvdsgcd  11707  gcddiv  11714  txcnp  12450  blssps  12606  blss  12607  tgqioo  12726
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