Metamath Proof Explorer

Theorem cbvoprab123davw

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

		Ref	Expression
	Hypothesis	cbvoprab123davw.1	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to (ψ \leftrightarrow χ)$
	Assertion	cbvoprab123davw	$⊢ φ \to \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{⟨⟨w, u⟩, v⟩ \| χ\}$

Proof

Step	Hyp	Ref	Expression
1		cbvoprab123davw.1	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to (ψ \leftrightarrow χ)$
2		simplr	$⊢ ((φ \land x = w) \land y = u) \to x = w$
3		simpr	$⊢ ((φ \land x = w) \land y = u) \to y = u$
4	2 3	opeq12d	$⊢ ((φ \land x = w) \land y = u) \to ⟨x, y⟩ = ⟨w, u⟩$
5	4	adantr	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to ⟨x, y⟩ = ⟨w, u⟩$
6		simpr	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to z = v$
7	5 6	opeq12d	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to ⟨⟨x, y⟩, z⟩ = ⟨⟨w, u⟩, v⟩$
8	7	eqeq2d	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to (t = ⟨⟨x, y⟩, z⟩ \leftrightarrow t = ⟨⟨w, u⟩, v⟩)$
9	8 1	anbi12d	$⊢ (((φ \land x = w) \land y = u) \land z = v) \to ((t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow (t = ⟨⟨w, u⟩, v⟩ \land χ))$
10	9	cbvexdvaw	$⊢ ((φ \land x = w) \land y = u) \to (\exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists v (t = ⟨⟨w, u⟩, v⟩ \land χ))$
11	10	cbvexdvaw	$⊢ (φ \land x = w) \to (\exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists u \exists v (t = ⟨⟨w, u⟩, v⟩ \land χ))$
12	11	cbvexdvaw	$⊢ φ \to (\exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ) \leftrightarrow \exists w \exists u \exists v (t = ⟨⟨w, u⟩, v⟩ \land χ))$
13	12	abbidv	$⊢ φ \to \{t \| \exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ)\} = \{t \| \exists w \exists u \exists v (t = ⟨⟨w, u⟩, v⟩ \land χ)\}$
14		df-oprab	$⊢ \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{t \| \exists x \exists y \exists z (t = ⟨⟨x, y⟩, z⟩ \land ψ)\}$
15		df-oprab	$⊢ \{⟨⟨w, u⟩, v⟩ \| χ\} = \{t \| \exists w \exists u \exists v (t = ⟨⟨w, u⟩, v⟩ \land χ)\}$
16	13 14 15	3eqtr4g	$⊢ φ \to \{⟨⟨x, y⟩, z⟩ \| ψ\} = \{⟨⟨w, u⟩, v⟩ \| χ\}$