Theorem 19.27

Description: Theorem 19.27 of Margaris p. 90. See 19.27v for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	19.27.1	$$⊢Ⅎx\psi$$
Assertion	19.27	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.27.1	$$⊢Ⅎx\psi$$
2		19.26	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall x\psi \right)$$
3	1	19.3	$$⊢\forall x\psi \leftrightarrow \psi$$
4	3	anbi2i	$$⊢\left(\forall x\phi \wedge \forall x\psi \right)\leftrightarrow \left(\forall x\phi \wedge \psi \right)$$
5	2 4	bitri	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \psi \right)$$