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Theorem rexlimdva2 2550
 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 361 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
32rexlimdv 2546 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  ∃wrex 2415 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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419  df-rex 2420 This theorem is referenced by:  ctssdclemn0  6988  ctssdc  6991  suplocexprlemru  7520  suplocexprlemloc  7522  suplocsrlemb  7607  ennnfonelemhom  11917  innei  12321  ivthinclemlr  12773  ivthinclemur  12775  limccnpcntop  12802  limccoap  12805
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