Theorem cbvoprab23vw 36453

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

Hypothesis

Ref	Expression
cbvoprab23vw.1	⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvoprab23vw	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒}

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣   𝜓,𝑤,𝑣   𝜒,𝑦,𝑧

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑤,𝑣)

Proof of Theorem cbvoprab23vw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 4832	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
2	1	adantr 480	. . . . . . . 8 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
3		simpr 484	. . . . . . . 8 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
4	2, 3	opeq12d 4839	. . . . . . 7 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩)
5	4	eqeq2d 2748	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩))
6		cbvoprab23vw.1	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))
7	5, 6	anbi12d 633	. . . . 5 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
8	7	cbvex2vw 2043	. . . 4 ⊢ (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒))
9	8	exbii 1850	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒))
10	9	abbii 2804	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
11		df-oprab 7372	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
12		df-oprab 7372	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
13	10, 11, 12	3eqtr4i 2770	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒}

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  ⟨cop 4588  {coprab 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-oprab 7372

This theorem is referenced by: (None)