Theorem aaanv

Description: Theorem *11.56 in WhiteheadRussell p. 165. Special case of aaan . (Contributed by Andrew Salmon, 24-May-2011)

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

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎy\phi$$
2		nfv	$$⊢Ⅎx\psi$$
3	1 2	aaan	$$⊢\forall x\forall y\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall y\psi \right)$$
4	3	bicomi	$$⊢\left(\forall x\phi \wedge \forall y\psi \right)\leftrightarrow \forall x\forall y\left(\phi \wedge \psi \right)$$