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Theorem bj-exlimg 34013
Description: The general form of the *exlim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg 34009. (Contributed by BJ, 9-Dec-2023.)
Ref Expression
bj-exlimg ((∃𝑥𝜑𝜓) → (∀𝑥(𝜒𝜑) → (∃𝑥𝜒𝜓)))

Proof of Theorem bj-exlimg
StepHypRef Expression
1 bj-sylget 34011 . 2 (∀𝑥(𝜒𝜑) → ((∃𝑥𝜑𝜓) → (∃𝑥𝜒𝜓)))
21com12 32 1 ((∃𝑥𝜑𝜓) → (∀𝑥(𝜒𝜑) → (∃𝑥𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-ex 1782
This theorem is referenced by:  bj-exlimd  34015  bj-nfimexal  34016  bj-subst  34112
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