Theorem aaan

Description: Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993) Avoid ax-10 . (Revised by Gino Giotto, 21-Nov-2024)

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

Proof

Step	Hyp	Ref	Expression
1		aaan.1	$$⊢Ⅎy\phi$$
2		aaan.2	$$⊢Ⅎx\psi$$
3		19.26-2	$$⊢\forall x\forall y\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\forall y\phi \wedge \forall x\forall y\psi \right)$$
4	1	19.3	$$⊢\forall y\phi \leftrightarrow \phi$$
5	4	albii	$$⊢\forall x\forall y\phi \leftrightarrow \forall x\phi$$
6		alcom	$$⊢\forall x\forall y\psi \leftrightarrow \forall y\forall x\psi$$
7	2	19.3	$$⊢\forall x\psi \leftrightarrow \psi$$
8	7	albii	$$⊢\forall y\forall x\psi \leftrightarrow \forall y\psi$$
9	6 8	bitri	$$⊢\forall x\forall y\psi \leftrightarrow \forall y\psi$$
10	5 9	anbi12i	$$⊢\left(\forall x\forall y\phi \wedge \forall x\forall y\psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall y\psi \right)$$
11	3 10	bitri	$$⊢\forall x\forall y\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall y\psi \right)$$