Theorem 2albi

Description: Theorem *11.33 in WhiteheadRussell p. 162. Theorem 19.15 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	2albi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		albi	⊢ ( ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑦 𝜑 ↔ ∀ 𝑦 𝜓 ) )
2	1	alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) → ∀ 𝑥 ( ∀ 𝑦 𝜑 ↔ ∀ 𝑦 𝜓 ) )
3		albi	⊢ ( ∀ 𝑥 ( ∀ 𝑦 𝜑 ↔ ∀ 𝑦 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜓 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ↔ 𝜓 ) → ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 𝜓 ) )