Theorem cbvoprab2vw

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

	Ref	Expression
Hypothesis	cbvoprab2vw.1	$$⊢y=w\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	cbvoprab2vw	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨⟨x,w⟩,z⟩|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		cbvoprab2vw.1	$$⊢y=w\to \left(\psi \leftrightarrow \chi \right)$$
2		opeq2	$$⊢y=w\to ⟨x,y⟩=⟨x,w⟩$$
3	2	opeq1d	$$⊢y=w\to ⟨⟨x,y⟩,z⟩=⟨⟨x,w⟩,z⟩$$
4	3	eqeq2d	$$⊢y=w\to \left(t=⟨⟨x,y⟩,z⟩\leftrightarrow t=⟨⟨x,w⟩,z⟩\right)$$
5	4 1	anbi12d	$$⊢y=w\to \left(\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \left(t=⟨⟨x,w⟩,z⟩\wedge \chi \right)\right)$$
6	5	exbidv	$$⊢y=w\to \left(\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists z\left(t=⟨⟨x,w⟩,z⟩\wedge \chi \right)\right)$$
7	6	cbvexvw	$$⊢\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists w\exists z\left(t=⟨⟨x,w⟩,z⟩\wedge \chi \right)$$
8	7	exbii	$$⊢\exists x\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists x\exists w\exists z\left(t=⟨⟨x,w⟩,z⟩\wedge \chi \right)$$
9	8	abbii	$$⊢\left\{t|\exists x\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\right\}=\left\{t|\exists x\exists w\exists z\left(t=⟨⟨x,w⟩,z⟩\wedge \chi \right)\right\}$$
10		df-oprab	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{t|\exists x\exists y\exists z\left(t=⟨⟨x,y⟩,z⟩\wedge \psi \right)\right\}$$
11		df-oprab	$$⊢\left\{⟨⟨x,w⟩,z⟩|\chi \right\}=\left\{t|\exists x\exists w\exists z\left(t=⟨⟨x,w⟩,z⟩\wedge \chi \right)\right\}$$
12	9 10 11	3eqtr4i	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨⟨x,w⟩,z⟩|\chi \right\}$$