Theorem sbcralg

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005) (Proof shortened by Andrew Salmon, 29-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		nfcv	$$⊢\underset{\_}{Ⅎ}yA$$
2		sbcralt	$$⊢\left(A\in V\wedge \underset{\_}{Ⅎ}yA\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in B\phi \leftrightarrow \forall y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
3	1 2	mpan2	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\in B\phi \leftrightarrow \forall y\in B\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$