Theorem nf5-1

Description: One direction of nf5 can be proved with a smaller footprint on axiom usage. (Contributed by Wolf Lammen, 16-Sep-2021)

	Ref	Expression
Assertion	nf5-1	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → Ⅎ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		exim	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑥 𝜑 ) )
2		hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
3	1 2	syl6	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
4	3	nfd	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → Ⅎ 𝑥 𝜑 )