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Metamath Proof Explorer

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	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow \forall x (x = A \to φ))$

Proof

Step	Hyp	Ref	Expression
1		df-sbc	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow A \in \{x \| φ\}$
2		elab6g	$⊢ A \in V \to (A \in \{x \| φ\} \leftrightarrow \forall x (x = A \to φ))$
3	1 2	bitrid	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow \forall x (x = A \to φ))$