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	⊢ ( ( ∃ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∃ 𝑥 𝜒 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-sylget	⊢ ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ( ∃ 𝑥 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜒 → 𝜓 ) ) )
2	1	com12	⊢ ( ( ∃ 𝑥 𝜑 → 𝜓 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∃ 𝑥 𝜒 → 𝜓 ) ) )