Theorem rabbida4

Description: Version of rabbidva2 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

	Ref	Expression
Hypotheses	rabbida4.nf	$$⊢Ⅎx\phi$$
	rabbida4.1	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
Assertion	rabbida4	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbida4.nf	$$⊢Ⅎx\phi$$
2		rabbida4.1	$$⊢\phi \to \left(\left(x\in A\wedge \psi \right)\leftrightarrow \left(x\in B\wedge \chi \right)\right)$$
3	1 2	abbid	$$⊢\phi \to \left\{x|\left(x\in A\wedge \psi \right)\right\}=\left\{x|\left(x\in B\wedge \chi \right)\right\}$$
4		df-rab	$$⊢\left\{x\in A|\psi \right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
5		df-rab	$$⊢\left\{x\in B|\chi \right\}=\left\{x|\left(x\in B\wedge \chi \right)\right\}$$
6	3 4 5	3eqtr4g	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in B|\chi \right\}$$