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Theorem rexlimdva2 39856
 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 631 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
32rexlimdv 3168 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∃wrex 3051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055  df-rex 3056 This theorem is referenced by:  supminfxr  40210  infrpgernmpt  40211  limsupresxr  40519  liminfresxr  40520  liminflelimsuplem  40528  limsupgtlem  40530  liminfvalxr  40536  liminfreuzlem  40555  cnrefiisplem  40576  xlimmnfvlem2  40580  xlimpnfvlem2  40584  smfliminflem  41560
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