Theorem sbc6g

Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Proof shortened by SN, 5-Oct-2024)

	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		df-sbc	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow A\in \left\{x|\phi \right\}$$
2		elab6g	$$⊢A\in V\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$
3	1 2	syl5bb	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \forall x\left(x=A\to \phi \right)\right)$$