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Metamath Proof Explorer

Theorem bnj538

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020)

		Ref	Expression
	Hypothesis	bnj538.1	$⊢ A \in V$
	Assertion	bnj538	$⊢ \underset{˙}{[} A / y \underset{˙}{]} \forall x \in B φ \leftrightarrow \forall x \in B \underset{˙}{[} A / y \underset{˙}{]} φ$

Proof

Step	Hyp	Ref	Expression
1		bnj538.1	$⊢ A \in V$
2		sbcralg	$⊢ A \in V \to (\underset{˙}{[} A / y \underset{˙}{]} \forall x \in B φ \leftrightarrow \forall x \in B \underset{˙}{[} A / y \underset{˙}{]} φ)$
3	1 2	ax-mp	$⊢ \underset{˙}{[} A / y \underset{˙}{]} \forall x \in B φ \leftrightarrow \forall x \in B \underset{˙}{[} A / y \underset{˙}{]} φ$