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	$$⊢A\in \mathrm{V}$$
Assertion	bnj538	$$⊢\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\forall x\in B\phi \leftrightarrow \forall x\in B\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$

Proof

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