Theorem aaanOLD

Description: Obsolete version of aaan as of 21-Nov-2024. (Contributed by NM, 12-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	aaan.1	$$⊢Ⅎy\phi$$
	aaan.2	$$⊢Ⅎx\psi$$
Assertion	aaanOLD	$$⊢\forall x\forall y\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		aaan.1	$$⊢Ⅎy\phi$$
2		aaan.2	$$⊢Ⅎx\psi$$
3	1	19.28	$$⊢\forall y\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall y\psi \right)$$
4	3	albii	$$⊢\forall x\forall y\left(\phi \wedge \psi \right)\leftrightarrow \forall x\left(\phi \wedge \forall y\psi \right)$$
5	2	nfal	$$⊢Ⅎx\forall y\psi$$
6	5	19.27	$$⊢\forall x\left(\phi \wedge \forall y\psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall y\psi \right)$$
7	4 6	bitri	$$⊢\forall x\forall y\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall y\psi \right)$$