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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 ⊢ φ → ∀ x ψ → ∀ x ψ → χ → φ → ∀ x χ

Proof

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