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Theorem cbvoprab3 7234
Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvoprab3.1 𝑤𝜑
cbvoprab3.2 𝑧𝜓
cbvoprab3.3 (𝑧 = 𝑤 → (𝜑𝜓))
Assertion
Ref Expression
cbvoprab3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvoprab3
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1906 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab3.1 . . . . . 6 𝑤𝜑
31, 2nfan 1891 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 2334 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 2334 . . 3 𝑤𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6 nfv 1906 . . . . . 6 𝑧 𝑣 = ⟨𝑥, 𝑦
7 cbvoprab3.2 . . . . . 6 𝑧𝜓
86, 7nfan 1891 . . . . 5 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
98nfex 2334 . . . 4 𝑧𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
109nfex 2334 . . 3 𝑧𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11 cbvoprab3.3 . . . . 5 (𝑧 = 𝑤 → (𝜑𝜓))
1211anbi2d 628 . . . 4 (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13122exbidv 1916 . . 3 (𝑧 = 𝑤 → (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
145, 10, 13cbvopab2 5132 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15 dfoprab2 7201 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16 dfoprab2 7201 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1714, 15, 163eqtr4i 2851 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wnf 1775  cop 4563  {copab 5119  {coprab 7146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-oprab 7149
This theorem is referenced by:  cbvoprab3v  7235  tposoprab  7917  erovlem  8382
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