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 φ x ψ x ψ χ φ x χ


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