Theorem 19.28vv

Description: Theorem *11.47 in WhiteheadRussell p. 164. Theorem 19.28 of Margaris p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	19.28vv	$$⊢\forall x\forall y\left(\psi \wedge \phi \right)\leftrightarrow \left(\psi \wedge \forall x\forall y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.28v	$$⊢\forall y\left(\psi \wedge \phi \right)\leftrightarrow \left(\psi \wedge \forall y\phi \right)$$
2	1	albii	$$⊢\forall x\forall y\left(\psi \wedge \phi \right)\leftrightarrow \forall x\left(\psi \wedge \forall y\phi \right)$$
3		19.28v	$$⊢\forall x\left(\psi \wedge \forall y\phi \right)\leftrightarrow \left(\psi \wedge \forall x\forall y\phi \right)$$
4	2 3	bitri	$$⊢\forall x\forall y\left(\psi \wedge \phi \right)\leftrightarrow \left(\psi \wedge \forall x\forall y\phi \right)$$