Theorem albi

Description: Theorem 19.15 of Margaris p. 90. (Contributed by NM, 24-Jan-1993)

	Ref	Expression
Assertion	albi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		biimp	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
2	1	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 𝜑 → ∀ 𝑥 𝜓 ) )
3		biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
4	3	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜑 ) )
5	2 4	impbid	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 ) )