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Theorem cbvoprab3 7492
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 1909 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab3.1 . . . . . 6 𝑤𝜑
31, 2nfan 1894 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 2309 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 2309 . . 3 𝑤𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6 nfv 1909 . . . . . 6 𝑧 𝑣 = ⟨𝑥, 𝑦
7 cbvoprab3.2 . . . . . 6 𝑧𝜓
86, 7nfan 1894 . . . . 5 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
98nfex 2309 . . . 4 𝑧𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
109nfex 2309 . . 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 1919 . . 3 (𝑧 = 𝑤 → (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
145, 10, 13cbvopab2 5215 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15 dfoprab2 7459 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16 dfoprab2 7459 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1714, 15, 163eqtr4i 2762 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wnf 1777  cop 4626  {copab 5200  {coprab 7402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-opab 5201  df-oprab 7405
This theorem is referenced by:  cbvoprab3v  7493  tposoprab  8242  erovlem  8802
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