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Theorem cbvoprab12 5813
Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
cbvoprab12.1 𝑤𝜑
cbvoprab12.2 𝑣𝜑
cbvoprab12.3 𝑥𝜓
cbvoprab12.4 𝑦𝜓
cbvoprab12.5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
Assertion
Ref Expression
cbvoprab12 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem cbvoprab12
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 nfv 1493 . . . . 5 𝑤 𝑢 = ⟨𝑥, 𝑦
2 cbvoprab12.1 . . . . 5 𝑤𝜑
31, 2nfan 1529 . . . 4 𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1493 . . . . 5 𝑣 𝑢 = ⟨𝑥, 𝑦
5 cbvoprab12.2 . . . . 5 𝑣𝜑
64, 5nfan 1529 . . . 4 𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1493 . . . . 5 𝑥 𝑢 = ⟨𝑤, 𝑣
8 cbvoprab12.3 . . . . 5 𝑥𝜓
97, 8nfan 1529 . . . 4 𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10 nfv 1493 . . . . 5 𝑦 𝑢 = ⟨𝑤, 𝑣
11 cbvoprab12.4 . . . . 5 𝑦𝜓
1210, 11nfan 1529 . . . 4 𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13 opeq12 3677 . . . . . 6 ((𝑥 = 𝑤𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
1413eqeq2d 2129 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝑢 = ⟨𝑥, 𝑦⟩ ↔ 𝑢 = ⟨𝑤, 𝑣⟩))
15 cbvoprab12.5 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
1614, 15anbi12d 464 . . . 4 ((𝑥 = 𝑤𝑦 = 𝑣) → ((𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2 1874 . . 3 (∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
1817opabbii 3965 . 2 {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19 dfoprab2 5786 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 dfoprab2 5786 . 2 {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2148 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wnf 1421  wex 1453  cop 3500  {copab 3958  {coprab 5743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-oprab 5746
This theorem is referenced by:  cbvoprab12v  5814  cbvmpox  5817  dfoprab4f  6059  fmpox  6066  tposoprab  6145
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