Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvopab Structured version   Visualization version   GIF version

Theorem cbvopab 5101
 Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1 𝑧𝜑
cbvopab.2 𝑤𝜑
cbvopab.3 𝑥𝜓
cbvopab.4 𝑦𝜓
cbvopab.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvopab {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvopab
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . 5 𝑧 𝑣 = ⟨𝑥, 𝑦
2 cbvopab.1 . . . . 5 𝑧𝜑
31, 2nfan 1900 . . . 4 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1915 . . . . 5 𝑤 𝑣 = ⟨𝑥, 𝑦
5 cbvopab.2 . . . . 5 𝑤𝜑
64, 5nfan 1900 . . . 4 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1915 . . . . 5 𝑥 𝑣 = ⟨𝑧, 𝑤
8 cbvopab.3 . . . . 5 𝑥𝜓
97, 8nfan 1900 . . . 4 𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
10 nfv 1915 . . . . 5 𝑦 𝑣 = ⟨𝑧, 𝑤
11 cbvopab.4 . . . . 5 𝑦𝜓
1210, 11nfan 1900 . . . 4 𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
13 opeq12 4767 . . . . . 6 ((𝑥 = 𝑧𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
1413eqeq2d 2809 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑧, 𝑤⟩))
15 cbvopab.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1614, 15anbi12d 633 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2v 2354 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
1817abbii 2863 . 2 {𝑣 ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
19 df-opab 5093 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 df-opab 5093 . 2 {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2831 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  Ⅎwnf 1785  {cab 2776  ⟨cop 4531  {copab 5092 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093 This theorem is referenced by:  cbvopabv  5102  dfrel4  6015  bj-opabco  34619  aomclem8  40020
 Copyright terms: Public domain W3C validator