Theorem rspcdv

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-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	rspcdv	$$⊢\phi \to \left(\forall x\in B\psi \to \chi \right)$$

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	biimpd	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \to \chi \right)$$
4	1 3	rspcimdv	$$⊢\phi \to \left(\forall x\in B\psi \to \chi \right)$$