Theorem pm11.12

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

	Ref	Expression
Assertion	pm11.12	$$⊢\forall x\forall y\left(\phi \vee \psi \right)\to \left(\phi \vee \forall x\forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm10.12	$$⊢\forall y\left(\phi \vee \psi \right)\to \left(\phi \vee \forall y\psi \right)$$
2	1	alimi	$$⊢\forall x\forall y\left(\phi \vee \psi \right)\to \forall x\left(\phi \vee \forall y\psi \right)$$
3		pm10.12	$$⊢\forall x\left(\phi \vee \forall y\psi \right)\to \left(\phi \vee \forall x\forall y\psi \right)$$
4	2 3	syl	$$⊢\forall x\forall y\left(\phi \vee \psi \right)\to \left(\phi \vee \forall x\forall y\psi \right)$$