Theorem 19.40-2

Description: Theorem *11.42 in WhiteheadRussell p. 163. Theorem 19.40 of Margaris p. 90 with two quantifiers. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	19.40-2	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\to \left(\exists x\exists y\phi \wedge \exists x\exists y\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.40	$$⊢\exists y\left(\phi \wedge \psi \right)\to \left(\exists y\phi \wedge \exists y\psi \right)$$
2	1	eximi	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\to \exists x\left(\exists y\phi \wedge \exists y\psi \right)$$
3		19.40	$$⊢\exists x\left(\exists y\phi \wedge \exists y\psi \right)\to \left(\exists x\exists y\phi \wedge \exists x\exists y\psi \right)$$
4	2 3	syl	$$⊢\exists x\exists y\left(\phi \wedge \psi \right)\to \left(\exists x\exists y\phi \wedge \exists x\exists y\psi \right)$$