Theorem cbvoprab1vw

Description: Change the first bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypothesis	cbvoprab1vw.1	$$⊢x=w\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvoprab1vw	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨⟨w,y⟩,z⟩|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvoprab1vw.1	$$⊢x=w\to \left(\psi \leftrightarrow \chi \right)$$
2		opeq1	$$⊢x=w\to ⟨x,y⟩=⟨w,y⟩$$
3	2	opeq1d	$$⊢x=w\to ⟨⟨x,y⟩,z⟩=⟨⟨w,y⟩,z⟩$$
4	3	eqeq2d	$$⊢x=w\to \left(t=⟨⟨x,y⟩,z⟩\leftrightarrow t=⟨⟨w,y⟩,z⟩\right)$$
5	4 1	anbi12d	$$⊢x=w\to \left(\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \left(t=⟨⟨w,y⟩,z⟩\wedge \chi \right)\right)$$
6	5	2exbidv	$$⊢x=w\to \left(\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists y\exists z\left(t=⟨⟨w,y⟩,z⟩\wedge \chi \right)\right)$$
7	6	cbvexvw	$$⊢\exists x\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists w\exists y\exists z\left(t=⟨⟨w,y⟩,z⟩\wedge \chi \right)$$
8	7	abbii	$$⊢\left\{t|\exists x\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\right\}=\left\{t|\exists w\exists y\exists z\left(t=⟨⟨w,y⟩,z⟩\wedge \chi \right)\right\}$$
9		df-oprab	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{t|\exists x\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\right\}$$
10		df-oprab	$$⊢\left\{⟨⟨w,y⟩,z⟩|\chi \right\}=\left\{t|\exists w\exists y\exists z\left(t=⟨⟨w,y⟩,z⟩\wedge \chi \right)\right\}$$
11	8 9 10	3eqtr4i	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨⟨w,y⟩,z⟩|\chi \right\}$$