Theorem aaanv

Description: Theorem *11.56 in WhiteheadRussell p. 165. Special case of aaan . (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	aaanv	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑦 𝜓 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑦 𝜑
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	1 2	aaan	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∀ 𝑥 𝜑 ∧ ∀ 𝑦 𝜓 ) )
4	3	bicomi	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑦 𝜓 ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∧ 𝜓 ) )