Theorem rabbia2

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	rabbia2.1	$$⊢\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)$$
Assertion	rabbia2	$$⊢\left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbia2.1	$$⊢\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)$$
2	1	a1i	$$⊢\top \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
3	2	rabbidva2	$$⊢\top \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$
4	3	mptru	$$⊢\left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$