Theorem bj-spimenfa

Description: An existential generalization result: if ph holds and implies ps for at least one value of x , and if furthermore x is A. -weakly nonfree in ph , then ps holds for at least one value of x . (Contributed by BJ, 3-Apr-2026) Proof should not use 19.35 . (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-spimenfa	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-eximcom	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		imim1	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )
3	1 2	syl5	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )