Theorem r19.40

Description: Restricted quantifier version of Theorem 19.40 of Margaris p. 90. (Contributed by NM, 2-Apr-2004)

	Ref	Expression
Assertion	r19.40	$$⊢\exists x\in A\left(\phi \wedge \psi \right)\to \left(\exists x\in A\phi \wedge \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(\phi \wedge \psi \right)\to \phi$$
2	1	reximi	$$⊢\exists x\in A\left(\phi \wedge \psi \right)\to \exists x\in A\phi$$
3		simpr	$$⊢\left(\phi \wedge \psi \right)\to \psi$$
4	3	reximi	$$⊢\exists x\in A\left(\phi \wedge \psi \right)\to \exists x\in A\psi$$
5	2 4	jca	$$⊢\exists x\in A\left(\phi \wedge \psi \right)\to \left(\exists x\in A\phi \wedge \exists x\in A\psi \right)$$