Theorem r19.26

Description: Restricted quantifier version of 19.26 . (Contributed by NM, 28-Jan-1997) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	r19.26	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
2	1	ralimi	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\to \forall x\in A\phi$$
3		simpr	$$⊢\left(\phi \wedge \psi \right)\to \psi$$
4	3	ralimi	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\to \forall x\in A\psi$$
5	2 4	jca	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\to \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$
6		pm3.2	$$⊢\phi \to \left(\psi \to \left(\phi \wedge \psi \right)\right)$$
7	6	ral2imi	$$⊢\forall x\in A\phi \to \left(\forall x\in A\psi \to \forall x\in A\left(\phi \wedge \psi \right)\right)$$
8	7	imp	$$⊢\left(\forall x\in A\phi \wedge \forall x\in A\psi \right)\to \forall x\in A\left(\phi \wedge \psi \right)$$
9	5 8	impbii	$$⊢\forall x\in A\left(\phi \wedge \psi \right)\leftrightarrow \left(\forall x\in A\phi \wedge \forall x\in A\psi \right)$$