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Theorem rexlimd3 41411
Description: * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
rexlimd3.1 𝑥𝜑
rexlimd3.2 𝑥𝜒
rexlimd3.3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
Assertion
Ref Expression
rexlimd3 (𝜑 → (∃𝑥𝐴 𝜓𝜒))

Proof of Theorem rexlimd3
StepHypRef Expression
1 rexlimd3.1 . 2 𝑥𝜑
2 rexlimd3.2 . 2 𝑥𝜒
3 rexlimd3.3 . . 3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43exp31 422 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
51, 2, 4rexlimd 3317 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wnf 1780  wcel 2110  wrex 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-nf 1781  df-ral 3143  df-rex 3144
This theorem is referenced by:  dffo3f  41436
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