Theorem 19.29r

Description: Variation of 19.29 . (Contributed by NM, 18-Aug-1993) (Proof shortened by Wolf Lammen, 12-Nov-2020)

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

Proof

Step	Hyp	Ref	Expression
1		pm3.21	$$⊢\psi \to \left(\phi \to \left(\phi \wedge \psi \right)\right)$$
2	1	aleximi	$$⊢\forall x\psi \to \left(\exists x\phi \to \exists x\left(\phi \wedge \psi \right)\right)$$
3	2	impcom	$$⊢\left(\exists x\phi \wedge \forall x\psi \right)\to \exists x\left(\phi \wedge \psi \right)$$