cbvoprab1davw

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

		Ref	Expression
	Hypothesis	cbvoprab1davw.1	$⊢ (φ \land x = w) \to (ψ \leftrightarrow χ)$
	Assertion	cbvoprab1davw	$⊢ φ \to \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{⟨⟨w, y⟩, z⟩ \| χ\}$

Step	Hyp	Ref	Expression
1		cbvoprab1davw.1	$⊢ (φ \land x = w) \to (ψ \leftrightarrow χ)$
2		opeq1	$⊢ x = w \to ⟨x, y⟩ = ⟨w, y⟩$
3	2	adantl	$⊢ (φ \land x = w) \to ⟨x, y⟩ = ⟨w, y⟩$
4	3	opeq1d	$⊢ (φ \land x = w) \to ⟨⟨x, y⟩, z⟩ = ⟨⟨w, y⟩, z⟩$
5	4	eqeq2d	$⊢ (φ \land x = w) \to (t = ⟨⟨x, y⟩, z⟩ \leftrightarrow t = ⟨⟨w, y⟩, z⟩)$
6	5 1	anbi12d	$⊢ (φ \land x = w) \to ((t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow (t = ⟨⟨w, y⟩, z⟩ \land χ))$
7	6	2exbidv	$⊢ (φ \land x = w) \to (\exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists y \exists z (t = ⟨⟨w, y⟩, z⟩ \land χ))$
8	7	cbvexdvaw	$⊢ φ \to (\exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists w \exists y \exists z (t = ⟨⟨w, y⟩, z⟩ \land χ))$
9	8	abbidv	$⊢ φ \to \{t \| \exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ)\} = \{t \| \exists w \exists y \exists z (t = ⟨⟨w, y⟩, z⟩ \land χ)\}$
10		df-oprab	$⊢ \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{t \| \exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ)\}$
11		df-oprab	$⊢ \{⟨⟨w, y⟩, z⟩ \| χ\} = \{t \| \exists w \exists y \exists z (t = ⟨⟨w, y⟩, z⟩ \land χ)\}$
12	9 10 11	3eqtr4g	$⊢ φ \to \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{⟨⟨w, y⟩, z⟩ \| χ\}$

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