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Metamath Proof Explorer

Theorem 19.37imv

Description: One direction of 19.37v that can be proven without ax-6 . (Contributed by Rohan Ridenour, 16-Apr-2022)

		Ref	Expression
	Assertion	19.37imv	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) )

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		19.35	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	2	biimpi	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( ∀ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
4	1 3	syl5	⊢ ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ( 𝜑 → ∃ 𝑥 𝜓 ) )