Theorem rspccva

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006) (Proof shortened by Andrew Salmon, 8-Jun-2011)

	Ref	Expression
Hypothesis	rspcv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rspccva	$$⊢\left(\forall x\in B\phi \wedge A\in B\right)\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		rspcv.1	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	rspcv	$$⊢A\in B\to \left(\forall x\in B\phi \to \psi \right)$$
3	2	impcom	$$⊢\left(\forall x\in B\phi \wedge A\in B\right)\to \psi$$