Theorem r19.29an

Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux, 29-Dec-2019) (Proof shortened by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Hypothesis	rexlimdva2.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
Assertion	r19.29an	$$⊢\left(\phi \wedge \exists x\in A\psi \right)\to \chi$$

Proof

Step	Hyp	Ref	Expression
1		rexlimdva2.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
2	1	rexlimdva2	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$
3	2	imp	$$⊢\left(\phi \wedge \exists x\in A\psi \right)\to \chi$$