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	$$⊢\forall x\forall y\left(\phi \vee \psi \right)\leftrightarrow \left(\forall x\forall y\phi \vee \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.31v	$$⊢\forall y\left(\phi \vee \psi \right)\leftrightarrow \left(\forall y\phi \vee \psi \right)$$
2	1	albii	$$⊢\forall x\forall y\left(\phi \vee \psi \right)\leftrightarrow \forall x\left(\forall y\phi \vee \psi \right)$$
3		19.31v	$$⊢\forall x\left(\forall y\phi \vee \psi \right)\leftrightarrow \left(\forall x\forall y\phi \vee \psi \right)$$
4	2 3	bitri	$$⊢\forall x\forall y\left(\phi \vee \psi \right)\leftrightarrow \left(\forall x\forall y\phi \vee \psi \right)$$