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Theorem rexlimd3 39649
 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 629 . 2 (𝜑 → (𝑥𝐴 → (𝜓𝜒)))
51, 2, 4rexlimd 3055 1 (𝜑 → (∃𝑥𝐴 𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  Ⅎwnf 1748   ∈ wcel 2030  ∃wrex 2942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-ral 2946  df-rex 2947 This theorem is referenced by: (None)
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