Theorem rabbida2

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	rabbida2.1	$$⊢Ⅎx\phi$$
	rabbida2.2	$$⊢\phi \to A=B$$
	rabbida2.3	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	rabbida2	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbida2.1	$$⊢Ⅎx\phi$$
2		rabbida2.2	$$⊢\phi \to A=B$$
3		rabbida2.3	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
4	2	eleq2d	$$⊢\phi \to \left(x\in A\leftrightarrow x\in B\right)$$
5	4 3	anbi12d	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
6	1 5	abbid	$$⊢\phi \to \left\{x|\left(x\in A\wedge \psi \right)\right\}=\left\{x|\left(x\in B\wedge \chi \right)\right\}$$
7		df-rab	$$⊢\left\{x\in A|\psi \right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
8		df-rab	$$⊢\left\{x\in B|\chi \right\}=\left\{x|\left(x\in B\wedge \chi \right)\right\}$$
9	6 7 8	3eqtr4g	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$