Theorem rabbidva2

Description: Equivalent wff's yield equal restricted class abstractions. (Contributed by Thierry Arnoux, 4-Feb-2017)

	Ref	Expression
Hypothesis	rabbidva2.1	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
Assertion	rabbidva2	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbidva2.1	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
2	1	abbidv	$$⊢\phi \to \left\{x|\left(x\in A\wedge \psi \right)\right\}=\left\{x|\left(x\in B\wedge \chi \right)\right\}$$
3		df-rab	$$⊢\left\{x\in A|\psi \right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
4		df-rab	$$⊢\left\{x\in B|\chi \right\}=\left\{x|\left(x\in B\wedge \chi \right)\right\}$$
5	2 3 4	3eqtr4g	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$