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	$$⊢\left(\exists x\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\exists x\chi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-sylget	$$⊢\forall x\left(\chi \to \phi \right)\to \left(\left(\exists x\phi \to \psi \right)\to \left(\exists x\chi \to \psi \right)\right)$$
2	1	com12	$$⊢\left(\exists x\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\exists x\chi \to \psi \right)\right)$$