Theorem bj-spimnfe

Description: A universal specification result: if ph is true for all values of x and implies ps for at least one value, and if furthermore x is E. -weakly nonfree in ps , then ps follows. An intermediate result on the way to prove 19.36i , bj-19.36im , 19.36imv , spimfw ... (Contributed by BJ, 3-Apr-2026) Proof should not use 19.35 . (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-eximcom	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		imim2	⊢ ( ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )
3	1 2	syl5	⊢ ( ( ∃ 𝑥 𝜓 → 𝜓 ) → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) ) )