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Theorem rexlimdvaa 2588
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 373 . 2 ((𝜑𝑥𝐴) → (𝜓𝜒))
32rexlimdva 2587 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2141  wrex 2449
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454
This theorem is referenced by:  rexlimddv  2592  nnsucuniel  6474  omp1eomlem  7071  ctmlemr  7085  mulgt0sr  7740  axpre-suploclemres  7863  cnegex  8097  receuap  8587  rexanuz  10952  climcaucn  11314  fsumiun  11440  dvdsval2  11752  prmind2  12074  pcprmpw2  12286  pockthg  12309  tgcl  12858  neiint  12939  restopnb  12975  iscnp4  13012  blssexps  13223  blssex  13224  lgsne0  13733
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