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Theorem cbvoprab12 5606
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 1437 . . . . 5 𝑤 𝑢 = ⟨𝑥, 𝑦
2 cbvoprab12.1 . . . . 5 𝑤𝜑
31, 2nfan 1473 . . . 4 𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1437 . . . . 5 𝑣 𝑢 = ⟨𝑥, 𝑦
5 cbvoprab12.2 . . . . 5 𝑣𝜑
64, 5nfan 1473 . . . 4 𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1437 . . . . 5 𝑥 𝑢 = ⟨𝑤, 𝑣
8 cbvoprab12.3 . . . . 5 𝑥𝜓
97, 8nfan 1473 . . . 4 𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10 nfv 1437 . . . . 5 𝑦 𝑢 = ⟨𝑤, 𝑣
11 cbvoprab12.4 . . . . 5 𝑦𝜓
1210, 11nfan 1473 . . . 4 𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13 opeq12 3579 . . . . . 6 ((𝑥 = 𝑤𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
1413eqeq2d 2067 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝑢 = ⟨𝑥, 𝑦⟩ ↔ 𝑢 = ⟨𝑤, 𝑣⟩))
15 cbvoprab12.5 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
1614, 15anbi12d 450 . . . 4 ((𝑥 = 𝑤𝑦 = 𝑣) → ((𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2 1813 . . 3 (∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
1817opabbii 3852 . 2 {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19 dfoprab2 5580 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 dfoprab2 5580 . 2 {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2086 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wnf 1365  wex 1397  cop 3406  {copab 3845  {coprab 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-oprab 5544
This theorem is referenced by:  cbvoprab12v  5607  cbvmpt2x  5610  dfoprab4f  5847  fmpt2x  5854  tposoprab  5926
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