Theorem rspcedv

Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007) (Revised by Mario Carneiro, 4-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	$$⊢\phi \to A\in B$$
2		rspcdv.2	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	biimprd	$$⊢\left(\phi \wedge x=A\right)\to \left(\chi \to \psi \right)$$
4	1 3	rspcimedv	$$⊢\phi \to \left(\chi \to \exists x\in B\psi \right)$$