Theorem 19.27v

Description: Version of 19.27 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004)

	Ref	Expression
Assertion	19.27v	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.26	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall x\psi \right)$$
2		19.3v	$$⊢\forall x\psi \leftrightarrow \psi$$
3	2	anbi2i	$$⊢\left(\forall x\phi \wedge \forall x\psi \right)\leftrightarrow \left(\forall x\phi \wedge \psi \right)$$
4	1 3	bitri	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \psi \right)$$