Theorem 19.31vv

Description: Theorem *11.44 in WhiteheadRussell p. 163. Theorem 19.31 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	19.31vv	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∨ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.31v	⊢ ( ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑦 𝜑 ∨ 𝜓 ) )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) ↔ ∀ 𝑥 ( ∀ 𝑦 𝜑 ∨ 𝜓 ) )
3		19.31v	⊢ ( ∀ 𝑥 ( ∀ 𝑦 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∨ 𝜓 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) ↔ ( ∀ 𝑥 ∀ 𝑦 𝜑 ∨ 𝜓 ) )