Theorem abbi1dv

Description: Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994) (Proof shortened by Wolf Lammen, 16-Nov-2019)

	Ref	Expression
Hypothesis	abbi1dv.1	$$⊢\phi \to \left(\psi \leftrightarrow x\in A\right)$$
Assertion	abbi1dv	$$⊢\phi \to \left\{x|\psi \right\}=A$$

Proof

Step	Hyp	Ref	Expression
1		abbi1dv.1	$$⊢\phi \to \left(\psi \leftrightarrow x\in A\right)$$
2	1	bicomd	$$⊢\phi \to \left(x\in A\leftrightarrow \psi \right)$$
3	2	abbi2dv	$$⊢\phi \to A=\left\{x|\psi \right\}$$
4	3	eqcomd	$$⊢\phi \to \left\{x|\psi \right\}=A$$