Theorem rspcimedv

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		rspcimdv.1	$$⊢\phi \to A\in B$$
2		rspcimedv.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\chi \to \psi \right)$$
3	2	con3d	$$⊢\left(\phi \wedge x=A\right)\to \left(\neg \psi \to \neg \chi \right)$$
4	1 3	rspcimdv	$$⊢\phi \to \left(\forall x\in B\neg \psi \to \neg \chi \right)$$
5	4	con2d	$$⊢\phi \to \left(\chi \to \neg \forall x\in B\neg \psi \right)$$
6		dfrex2	$$⊢\exists x\in B\psi \leftrightarrow \neg \forall x\in B\neg \psi$$
7	5 6	syl6ibr	$$⊢\phi \to \left(\chi \to \exists x\in B\psi \right)$$