Theorem pm11.12

Description: Theorem *11.12 in WhiteheadRussell p. 159. (Contributed by Andrew Salmon, 17-Jun-2011)

	Ref	Expression
Assertion	pm11.12	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∀ 𝑥 ∀ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm10.12	⊢ ( ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∀ 𝑦 𝜓 ) )
2	1	alimi	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∨ ∀ 𝑦 𝜓 ) )
3		pm10.12	⊢ ( ∀ 𝑥 ( 𝜑 ∨ ∀ 𝑦 𝜓 ) → ( 𝜑 ∨ ∀ 𝑥 ∀ 𝑦 𝜓 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝜑 ∨ 𝜓 ) → ( 𝜑 ∨ ∀ 𝑥 ∀ 𝑦 𝜓 ) )