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Theorem rexlimdvva 2455
Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
Hypothesis
Ref Expression
rexlimdvva.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
rexlimdvva (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 rexlimdvva
StepHypRef Expression
1 rexlimdvva.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))
21ex 112 . 2 (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝜓𝜒)))
32rexlimdvv 2454 1 (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1407  wrex 2322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-4 1414  ax-17 1433  ax-ial 1441  ax-i5r 1442
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-ral 2326  df-rex 2327
This theorem is referenced by:  ovelrn  5674  f1o2ndf1  5874  eroveu  6225  eroprf  6227  genipv  6635  genpelvl  6638  genpelvu  6639  genprndl  6647  genprndu  6648  addlocpr  6662  addnqprlemrl  6683  addnqprlemru  6684  mulnqprlemrl  6699  mulnqprlemru  6700  ltsopr  6722  ltaddpr  6723  ltexprlemfl  6735  ltexprlemrl  6736  ltexprlemfu  6737  ltexprlemru  6738  cauappcvgprlemladdfu  6780  cauappcvgprlemladdfl  6781  caucvgprlemdisj  6800  caucvgprlemladdfu  6803  caucvgprprlemdisj  6828  apreap  7622  apreim  7638  apirr  7640  apsym  7641  apcotr  7642  apadd1  7643  apneg  7646  mulext1  7647  apti  7657  qapne  8641  qtri3or  9170  qbtwnzlemex  9177  rebtwn2z  9181  cjap  9698  climcn2  10024  dvds2lem  10083
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