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Metamath Proof Explorer

Theorem sbc5

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993) (Revised by Mario Carneiro, 12-Oct-2016) (Proof shortened by SN, 2-Sep-2024)

		Ref	Expression
	Assertion	sbc5	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow \exists x (x = A \land φ)$

Proof

Step	Hyp	Ref	Expression
1		df-sbc	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow A \in \{x \| φ\}$
2		clelab	$⊢ A \in \{x \| φ\} \leftrightarrow \exists x (x = A \land φ)$
3	1 2	bitri	$⊢ \underset{˙}{[} A / x \underset{˙}{]} φ \leftrightarrow \exists x (x = A \land φ)$