Theorem bnj1454

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

	Ref	Expression
Hypothesis	bnj1454.1	$$⊢A=\left\{x|\phi \right\}$$
Assertion	bnj1454	$$⊢B\in \mathrm{V}\to \left(B\in A\leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj1454.1	$$⊢A=\left\{x|\phi \right\}$$
2	1	eleq2i	$$⊢B\in A\leftrightarrow B\in \left\{x|\phi \right\}$$
3		df-sbc	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \leftrightarrow B\in \left\{x|\phi \right\}$$
4	3	a1i	$$⊢B\in \mathrm{V}\to \left(\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \leftrightarrow B\in \left\{x|\phi \right\}\right)$$
5	2 4	bitr4id	$$⊢B\in \mathrm{V}\to \left(B\in A\leftrightarrow \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$