Theorem bj-19.37im

Description: One direction of 19.37 from the same axioms as 19.37imv . (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.37im	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
2		bj-nnfa	⊢ ( Ⅎ' 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3	2	imim1d	⊢ ( Ⅎ' 𝑥 𝜑 → ( ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )
4	1 3	syl5bi	⊢ ( Ⅎ' 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) ) )