Theorem rexlimddv

Description: Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016)

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

Proof

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