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Theorem bj-alrimg 34009
Description: The general form of the *alrim* 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 an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg 34013. (Contributed by BJ, 9-Dec-2023.)
Ref Expression
bj-alrimg ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓𝜒) → (𝜑 → ∀𝑥𝜒)))

Proof of Theorem bj-alrimg
StepHypRef Expression
1 sylgt 1823 . 2 (∀𝑥(𝜓𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
21com12 32 1 ((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓𝜒) → (𝜑 → ∀𝑥𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-4 1811
This theorem is referenced by:  bj-alrimd  34010  bj-nfimexal  34016
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