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Metamath Proof Explorer

Theorem bj-exlimg

Description: The general form of the *exlim* 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 a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg . (Contributed by BJ, 9-Dec-2023)

		Ref	Expression
	Assertion	bj-exlimg	\|- ( ( E. x ph -> ps ) -> ( A. x ( ch -> ph ) -> ( E. x ch -> ps ) ) )

Proof

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