Theorem rabbiia

Description: Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999) (Proof shortened by Wolf Lammen, 12-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		rabbiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.32i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \psi \right)$$
3	2	rabbia2	$$⊢\left\{x\in A|\phi \right\}=\left\{x\in A|\psi \right\}$$