Theorem xfree2

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

	Ref	Expression
Assertion	xfree2	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		xfree	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) )
2		eximal	⊢ ( ( ∃ 𝑥 𝜑 → 𝜑 ) ↔ ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
3	2	albii	⊢ ( ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) ↔ ∀ 𝑥 ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
4	1 3	bitri	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑥 ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )