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Theorem cbvoprab12 5578
 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 1421 . . . . 5 𝑤 𝑢 = ⟨𝑥, 𝑦
2 cbvoprab12.1 . . . . 5 𝑤𝜑
31, 2nfan 1457 . . . 4 𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1421 . . . . 5 𝑣 𝑢 = ⟨𝑥, 𝑦
5 cbvoprab12.2 . . . . 5 𝑣𝜑
64, 5nfan 1457 . . . 4 𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1421 . . . . 5 𝑥 𝑢 = ⟨𝑤, 𝑣
8 cbvoprab12.3 . . . . 5 𝑥𝜓
97, 8nfan 1457 . . . 4 𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10 nfv 1421 . . . . 5 𝑦 𝑢 = ⟨𝑤, 𝑣
11 cbvoprab12.4 . . . . 5 𝑦𝜓
1210, 11nfan 1457 . . . 4 𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13 opeq12 3551 . . . . . 6 ((𝑥 = 𝑤𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
1413eqeq2d 2051 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝑢 = ⟨𝑥, 𝑦⟩ ↔ 𝑢 = ⟨𝑤, 𝑣⟩))
15 cbvoprab12.5 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
1614, 15anbi12d 442 . . . 4 ((𝑥 = 𝑤𝑦 = 𝑣) → ((𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2 1797 . . 3 (∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
1817opabbii 3824 . 2 {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19 dfoprab2 5552 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 dfoprab2 5552 . 2 {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2070 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  Ⅎwnf 1349  ∃wex 1381  ⟨cop 3378  {copab 3817  {coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516 This theorem is referenced by:  cbvoprab12v  5579  cbvmpt2x  5582  dfoprab4f  5819  fmpt2x  5826  tposoprab  5895
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