Theorem 19.28

Description: Theorem 19.28 of Margaris p. 90. See 19.28v for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993)

	Ref	Expression
Hypothesis	19.28.1	$$⊢Ⅎx\phi$$
Assertion	19.28	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.28.1	$$⊢Ⅎx\phi$$
2		19.26	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\phi \wedge \forall x\psi \right)$$
3	1	19.3	$$⊢\forall x\phi \leftrightarrow \phi$$
4	3	anbi1i	$$⊢\left(\forall x\phi \wedge \forall x\psi \right)\leftrightarrow \left(\phi \wedge \forall x\psi \right)$$
5	2 4	bitri	$$⊢\forall x\left(\phi \wedge \psi \right)\leftrightarrow \left(\phi \wedge \forall x\psi \right)$$