Theorem xfree

Description: A partial converse to 19.9t . (Contributed by Stefan Allan, 21-Dec-2008) (Revised by Mario Carneiro, 11-Dec-2016)

	Ref	Expression
Assertion	xfree	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nf5	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )
2		nf6	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) )
3	1 2	bitr3i	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) )