Theorem rabbida

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

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

Proof

Step	Hyp	Ref	Expression
1		rabbida.n	$$⊢Ⅎx\phi$$
2		rabbida.1	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow \chi \right)$$
3	2	ex	$$⊢\phi \to \left(x\in A\to \left(\psi \leftrightarrow \chi \right)\right)$$
4	1 3	ralrimi	$$⊢\phi \to \forall x\in A\left(\psi \leftrightarrow \chi \right)$$
5		rabbi	$$⊢\forall x\in A\left(\psi \leftrightarrow \chi \right)\leftrightarrow \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$
6	4 5	sylib	$$⊢\phi \to \left\{x\in A|\psi \right\}=\left\{x\in A|\chi \right\}$$