Theorem sbc6g

Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011)

	Ref	Expression
Assertion	sbc6g	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sbc5	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\left(x=A\wedge \phi \right)$$
2		alexeqg	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
3	1 2	bitr4id	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$