Theorem r19.29a

Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux, 22-Nov-2017) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Hypotheses	r19.29a.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
	r19.29a.2	$$⊢\phi \to \exists x\in A\psi$$
Assertion	r19.29a	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		r19.29a.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
2		r19.29a.2	$$⊢\phi \to \exists x\in A\psi$$
3	1	rexlimdva2	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$
4	2 3	mpd	$$⊢\phi \to \chi$$