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

Proof

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