Theorem 19.16

Description: Theorem 19.16 of Margaris p. 90. (Contributed by NM, 12-Mar-1993)

	Ref	Expression
Hypothesis	19.16.1	⊢ Ⅎ 𝑥 𝜑
Assertion	19.16	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.16.1	⊢ Ⅎ 𝑥 𝜑
2	1	19.3	⊢ ( ∀ 𝑥 𝜑 ↔ 𝜑 )
3		albi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 ) )
4	2 3	bitr3id	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( 𝜑 ↔ ∀ 𝑥 𝜓 ) )