Theorem cbvoprab23vw 36637

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 4840	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
2	1	adantr 485	. . . . . . . 8 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
3		simpr 489	. . . . . . . 8 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
4	2, 3	opeq12d 4847	. . . . . . 7 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩)
5	4	eqeq2d 2780	. . . . . 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 643	. . . . 5 ⊢ ((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
8	7	cbvex2vw 2068	. . . 4 ⊢ (∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒))
9	8	exbii 1875	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒))
10	9	abbii 2836	. 2 ⊢ {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
11		df-oprab 7412	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥∃𝑦∃𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
12		df-oprab 7412	. 2 ⊢ {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥∃𝑤∃𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
13	10, 11, 12	3eqtr4i 2802	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒}

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806  {cab 2747  ⟨cop 4597  {coprab 7409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-oprab 7412

This theorem is referenced by: (None)