Theorem abbidv

Description: Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993) Avoid ax-12 , based on an idea of Steven Nguyen. (Revised by Wolf Lammen, 6-May-2023)

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

Proof

Step	Hyp	Ref	Expression
1		abbidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
3		abbi1	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$
4	2 3	syl	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$