Theorem r19.29af2

Description: A commonly used pattern based on r19.29 . (Contributed by Thierry Arnoux, 17-Dec-2017) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	r19.29af2.p	$$⊢Ⅎx\phi$$
	r19.29af2.c	$$⊢Ⅎx\chi$$
	r19.29af2.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
	r19.29af2.2	$$⊢\phi \to \exists x\in A\psi$$
Assertion	r19.29af2	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		r19.29af2.p	$$⊢Ⅎx\phi$$
2		r19.29af2.c	$$⊢Ⅎx\chi$$
3		r19.29af2.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
4		r19.29af2.2	$$⊢\phi \to \exists x\in A\psi$$
5	3	exp31	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
6	1 2 5	rexlimd	$$⊢\phi \to \left(\exists x\in A\psi \to \chi \right)$$
7	4 6	mpd	$$⊢\phi \to \chi$$