cbvoprab2davw

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

		Ref	Expression
	Hypothesis	cbvoprab2davw.1	$⊢ (φ \land y = w) \to (ψ \leftrightarrow χ)$
	Assertion	cbvoprab2davw	$⊢ φ \to \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{⟨⟨x, w⟩, z⟩ \| χ\}$

Step	Hyp	Ref	Expression
1		cbvoprab2davw.1	$⊢ (φ \land y = w) \to (ψ \leftrightarrow χ)$
2		opeq2	$⊢ y = w \to ⟨x, y⟩ = ⟨x, w⟩$
3	2	adantl	$⊢ (φ \land y = w) \to ⟨x, y⟩ = ⟨x, w⟩$
4	3	opeq1d	$⊢ (φ \land y = w) \to ⟨⟨x, y⟩, z⟩ = ⟨⟨x, w⟩, z⟩$
5	4	eqeq2d	$⊢ (φ \land y = w) \to (t = ⟨⟨x, y⟩, z⟩ \leftrightarrow t = ⟨⟨x, w⟩, z⟩)$
6	5 1	anbi12d	$⊢ (φ \land y = w) \to ((t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow (t = ⟨⟨x, w⟩, z⟩ \land χ))$
7	6	exbidv	$⊢ (φ \land y = w) \to (\exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists z (t = ⟨⟨x, w⟩, z⟩ \land χ))$
8	7	cbvexdvaw	$⊢ φ \to (\exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists w \exists z (t = ⟨⟨x, w⟩, z⟩ \land χ))$
9	8	exbidv	$⊢ φ \to (\exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists x \exists w \exists z (t = ⟨⟨x, w⟩, z⟩ \land χ))$
10	9	abbidv	$⊢ φ \to \{t \| \exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ)\} = \{t \| \exists x \exists w \exists z (t = ⟨⟨x, w⟩, z⟩ \land χ)\}$
11		df-oprab	$⊢ \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{t \| \exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ)\}$
12		df-oprab	$⊢ \{⟨⟨x, w⟩, z⟩ \| χ\} = \{t \| \exists x \exists w \exists z (t = ⟨⟨x, w⟩, z⟩ \land χ)\}$
13	10 11 12	3eqtr4g	$⊢ φ \to \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{⟨⟨x, w⟩, z⟩ \| χ\}$

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