Theorem rspsbca

Description: Restricted quantifier version of Axiom 4 of Mendelson p. 69. (Contributed by NM, 14-Dec-2005)

	Ref	Expression
Assertion	rspsbca	$$⊢\left(A\in B\wedge \forall x\in B\phi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		rspsbc	$$⊢A\in B\to \left(\forall x\in B\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
2	1	imp	$$⊢\left(A\in B\wedge \forall x\in B\phi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$