Theorem r19.29

Description: Restricted quantifier version of 19.29 . See also r19.29r . (Contributed by NM, 31-Aug-1999) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	r19.29	$$⊢\left(\forall x\in A\phi \wedge \exists x\in A\psi \right)\to \exists x\in A\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm3.2	$$⊢\phi \to \left(\psi \to \left(\phi \wedge \psi \right)\right)$$
2	1	ralimi	$$⊢\forall x\in A\phi \to \forall x\in A\left(\psi \to \left(\phi \wedge \psi \right)\right)$$
3		rexim	$$⊢\forall x\in A\left(\psi \to \left(\phi \wedge \psi \right)\right)\to \left(\exists x\in A\psi \to \exists x\in A\left(\phi \wedge \psi \right)\right)$$
4	2 3	syl	$$⊢\forall x\in A\phi \to \left(\exists x\in A\psi \to \exists x\in A\left(\phi \wedge \psi \right)\right)$$
5	4	imp	$$⊢\left(\forall x\in A\phi \wedge \exists x\in A\psi \right)\to \exists x\in A\left(\phi \wedge \psi \right)$$