Theorem rabbidva

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003) (Proof shortened by SN, 3-Dec-2023)

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

Proof

Step	Hyp	Ref	Expression
1		rabbidva.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
2	1	pm5.32da	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in A\wedge \chi \right)\right)$$
3	2	rabbidva2	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$