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Theorem rexlimdvaa 2595
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
Hypothesis
Ref Expression
rexlimdvaa.1 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
Assertion
Ref Expression
rexlimdvaa (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimdvaa
StepHypRef Expression
1 rexlimdvaa.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝜓)) → 𝜒)
21expr 375 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rexlimdva 2594 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461
This theorem is referenced by:  rexlimddv  2599  nnsucuniel  6498  omp1eomlem  7095  ctmlemr  7109  mulgt0sr  7779  axpre-suploclemres  7902  cnegex  8137  receuap  8628  recapb  8630  rexanuz  10999  climcaucn  11361  fsumiun  11487  dvdsval2  11799  prmind2  12122  pcprmpw2  12334  pockthg  12357  dvdsrvald  13267  dvdsrd  13268  dvdsrex  13272  unitgrp  13290  tgcl  13649  neiint  13730  restopnb  13766  iscnp4  13803  blssexps  14014  blssex  14015  lgsne0  14524
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