Theorem r19.29r

Description: Restricted quantifier version of 19.29r ; variation of r19.29 . (Contributed by NM, 31-Aug-1999) (Proof shortened by Wolf Lammen, 29-Jun-2023)

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

Proof

Step	Hyp	Ref	Expression
1		iba	$$⊢\psi \to \left(\phi \leftrightarrow \left(\phi \wedge \psi \right)\right)$$
2	1	ralrexbid	$$⊢\forall x\in A\psi \to \left(\exists x\in A\phi \leftrightarrow \exists x\in A\left(\phi \wedge \psi \right)\right)$$
3	2	biimpac	$$⊢\left(\exists x\in A\phi \wedge \forall x\in A\psi \right)\to \exists x\in A\left(\phi \wedge \psi \right)$$