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Metamath Proof Explorer

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	\|- ( ( ph -> A. x ps ) -> ( A. x ( ps -> ch ) -> ( ph -> A. x ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		sylgt	\|- ( A. x ( ps -> ch ) -> ( ( ph -> A. x ps ) -> ( ph -> A. x ch ) ) )
2	1	com12	\|- ( ( ph -> A. x ps ) -> ( A. x ( ps -> ch ) -> ( ph -> A. x ch ) ) )