Users' Mathboxes Mathbox for Gino Giotto < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvoprab23vw Structured version   Visualization version   GIF version

Theorem cbvoprab23vw 36183
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 = 𝑤𝑧 = 𝑣) → (𝜓𝜒))
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.
StepHypRef Expression
1 opeq2 4881 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
21adantr 480 . . . . . . . 8 ((𝑦 = 𝑤𝑧 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
3 simpr 484 . . . . . . . 8 ((𝑦 = 𝑤𝑧 = 𝑣) → 𝑧 = 𝑣)
42, 3opeq12d 4888 . . . . . . 7 ((𝑦 = 𝑤𝑧 = 𝑣) → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩)
54eqeq2d 2744 . . . . . 6 ((𝑦 = 𝑤𝑧 = 𝑣) → (𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩))
6 cbvoprab23vw.1 . . . . . 6 ((𝑦 = 𝑤𝑧 = 𝑣) → (𝜓𝜒))
75, 6anbi12d 631 . . . . 5 ((𝑦 = 𝑤𝑧 = 𝑣) → ((𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ (𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)))
87cbvex2vw 2036 . . . 4 (∃𝑦𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑤𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒))
98exbii 1843 . . 3 (∃𝑥𝑦𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑥𝑤𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒))
109abbii 2805 . 2 {𝑡 ∣ ∃𝑥𝑦𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)} = {𝑡 ∣ ∃𝑥𝑤𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
11 df-oprab 7429 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {𝑡 ∣ ∃𝑥𝑦𝑧(𝑡 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜓)}
12 df-oprab 7429 . 2 {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒} = {𝑡 ∣ ∃𝑥𝑤𝑣(𝑡 = ⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∧ 𝜒)}
1310, 11, 123eqtr4i 2771 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wex 1774  {cab 2710  cop 4636  {coprab 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-oprab 7429
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator