Theorem 19.29x

Description: Variation of 19.29 with mixed quantification. (Contributed by NM, 11-Feb-2005)

	Ref	Expression
Assertion	19.29x	$$⊢\left(\exists x\forall y\phi \wedge \forall x\exists y\psi \right)\to \exists x\exists y\left(\phi \wedge \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.29r	$$⊢\left(\exists x\forall y\phi \wedge \forall x\exists y\psi \right)\to \exists x\left(\forall y\phi \wedge \exists y\psi \right)$$
2		19.29	$$⊢\left(\forall y\phi \wedge \exists y\psi \right)\to \exists y\left(\phi \wedge \psi \right)$$
3	2	eximi	$$⊢\exists x\left(\forall y\phi \wedge \exists y\psi \right)\to \exists x\exists y\left(\phi \wedge \psi \right)$$
4	1 3	syl	$$⊢\left(\exists x\forall y\phi \wedge \forall x\exists y\psi \right)\to \exists x\exists y\left(\phi \wedge \psi \right)$$