Theorem 19.36imv

Description: One direction of 19.36v that can be proven without ax-6 . (Contributed by Rohan Ridenour, 16-Apr-2022) (Proof shortened by Wolf Lammen, 22-Sep-2024)

	Ref	Expression
Assertion	19.36imv	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.27	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
2	1	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 𝜓 ) )
3		ax5e	⊢ ( ∃ 𝑥 𝜓 → 𝜓 )
4	2 3	syl6com	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → 𝜓 ) )