Theorem abbid

Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993) (Revised by Mario Carneiro, 7-Oct-2016) Avoid ax-10 and ax-11 . (Revised by Wolf Lammen, 6-May-2023)

	Ref	Expression
Hypotheses	abbid.1	$$⊢Ⅎx\phi$$
	abbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	abbid	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		abbid.1	$$⊢Ⅎx\phi$$
2		abbid.2	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
3	1 2	alrimi	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
4		abbi1	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$
5	3 4	syl	$$⊢\phi \to \left\{x|\psi \right\}=\left\{x|\chi \right\}$$