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Theorem rexlimdva2 2577
Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
rexlimdva2.1 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
rexlimdva2 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Distinct variable groups:   𝜒,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimdva2
StepHypRef Expression
1 rexlimdva2.1 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
21exp31 362 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
32rexlimdv 2573 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2128  wrex 2436
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440  df-rex 2441
This theorem is referenced by:  ctssdclemn0  7056  ctssdc  7059  suplocexprlemru  7641  suplocexprlemloc  7643  suplocsrlemb  7728  ennnfonelemhom  12214  innei  12633  ivthinclemlr  13085  ivthinclemur  13087  limccnpcntop  13114  limccoap  13117
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