Theorem r19.27vOLD

Description: Obsolete version of r19.27v as of 17-Jun-2023. (Contributed by NM, 3-Jun-2004) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	r19.27vOLD	$$⊢\left(\forall x\in A\phi \wedge \psi \right)\to \forall x\in A\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$$⊢\psi \to \left(x\in A\to \psi \right)$$
2	1	ralrimiv	$$⊢\psi \to \forall x\in A\psi$$
3	2	anim2i	$$⊢\left(\forall x\in A\phi \wedge \psi \right)\to \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
4		r19.26	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
5	3 4	sylibr	$$⊢\left(\forall x\in A\phi \wedge \psi \right)\to \forall x\in A\left(\phi \wedge \psi \right)$$