Theorem r19.29af

Description: A commonly used pattern based on r19.29 . See r19.29a , r19.29an for a variant when x is disjoint from ph . (Contributed by Thierry Arnoux, 29-Nov-2017)

	Ref	Expression
Hypotheses	r19.29af.0	$$⊢Ⅎx\phi$$
	r19.29af.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
	r19.29af.2	$$⊢\phi \to \exists x\in A\psi$$
Assertion	r19.29af	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		r19.29af.0	$$⊢Ⅎx\phi$$
2		r19.29af.1	$$⊢\left(\left(\phi \wedge x\in A\right)\wedge \psi \right)\to \chi$$
3		r19.29af.2	$$⊢\phi \to \exists x\in A\psi$$
4		nfv	$$⊢Ⅎx\chi$$
5	1 4 2 3	r19.29af2	$$⊢\phi \to \chi$$