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Theorem cbvoprab3 5611
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 1462 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab3.1 . . . . . 6 𝑤𝜑
31, 2nfan 1498 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 1569 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 1569 . . 3 𝑤𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6 nfv 1462 . . . . . 6 𝑧 𝑣 = ⟨𝑥, 𝑦
7 cbvoprab3.2 . . . . . 6 𝑧𝜓
86, 7nfan 1498 . . . . 5 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
98nfex 1569 . . . 4 𝑧𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
109nfex 1569 . . 3 𝑧𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11 cbvoprab3.3 . . . . 5 (𝑧 = 𝑤 → (𝜑𝜓))
1211anbi2d 452 . . . 4 (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13122exbidv 1790 . . 3 (𝑧 = 𝑤 → (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
145, 10, 13cbvopab2 3860 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15 dfoprab2 5583 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16 dfoprab2 5583 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1714, 15, 163eqtr4i 2112 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wnf 1390  wex 1422  cop 3409  {copab 3846  {coprab 5544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848  df-oprab 5547
This theorem is referenced by:  cbvoprab3v  5612  tposoprab  5929  erovlem  6264
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