Theorem rspcdv2

Description: Restricted specialization, using implicit substitution. (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	rspcdv2.1	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
	rspcdv2.2	$$⊢\phi \to A\in B$$
	rspcdv2.3	$$⊢\phi \to \forall x\in B\psi$$
Assertion	rspcdv2	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		rspcdv2.1	$$⊢\left(\phi \wedge x=A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		rspcdv2.2	$$⊢\phi \to A\in B$$
3		rspcdv2.3	$$⊢\phi \to \forall x\in B\psi$$
4	2 1	rspcdv	$$⊢\phi \to \left(\forall x\in B\psi \to \chi \right)$$
5	3 4	mpd	$$⊢\phi \to \chi$$