Theorem bj-alrimg

Description: The general form of the *alrim* family of theorems: if ph is substituted for ps , then the antecedent expresses a form of nonfreeness of x in ph , 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 . (Contributed by BJ, 9-Dec-2023)

	Ref	Expression
Assertion	bj-alrimg	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( 𝜑 → ∀ 𝑥 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sylgt	⊢ ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( 𝜑 → ∀ 𝑥 𝜒 ) ) )
2	1	com12	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝜓 → 𝜒 ) → ( 𝜑 → ∀ 𝑥 𝜒 ) ) )