Theorem cbvrabv

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999) Require x , y be disjoint to avoid ax-11 and ax-13 . (Revised by Steven Nguyen, 4-Dec-2022)

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

Proof

Step	Hyp	Ref	Expression
1		cbvrabv.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		eleq1w	$$⊢x=y\to \left(x\in A\leftrightarrow y\in A\right)$$
3	2 1	anbi12d	$$⊢x=y\to \left(\left(x\in A\wedge \phi \right)\leftrightarrow \left(y\in A\wedge \psi \right)\right)$$
4	3	cbvabv	$$⊢\left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{y|\left(y\in A\wedge \psi \right)\right\}$$
5		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
6		df-rab	$$⊢\left\{y\in A|\psi \right\}=\left\{y|\left(y\in A\wedge \psi \right)\right\}$$
7	4 5 6	3eqtr4i	$$⊢\left\{x\in A|\phi \right\}=\left\{y\in A|\psi \right\}$$