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Theorem cbvoprab12 6726
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 1842 . . . . 5 𝑤 𝑢 = ⟨𝑥, 𝑦
2 cbvoprab12.1 . . . . 5 𝑤𝜑
31, 2nfan 1827 . . . 4 𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1842 . . . . 5 𝑣 𝑢 = ⟨𝑥, 𝑦
5 cbvoprab12.2 . . . . 5 𝑣𝜑
64, 5nfan 1827 . . . 4 𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1842 . . . . 5 𝑥 𝑢 = ⟨𝑤, 𝑣
8 cbvoprab12.3 . . . . 5 𝑥𝜓
97, 8nfan 1827 . . . 4 𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10 nfv 1842 . . . . 5 𝑦 𝑢 = ⟨𝑤, 𝑣
11 cbvoprab12.4 . . . . 5 𝑦𝜓
1210, 11nfan 1827 . . . 4 𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13 opeq12 4402 . . . . . 6 ((𝑥 = 𝑤𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
1413eqeq2d 2631 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝑢 = ⟨𝑥, 𝑦⟩ ↔ 𝑢 = ⟨𝑤, 𝑣⟩))
15 cbvoprab12.5 . . . . 5 ((𝑥 = 𝑤𝑦 = 𝑣) → (𝜑𝜓))
1614, 15anbi12d 747 . . . 4 ((𝑥 = 𝑤𝑦 = 𝑣) → ((𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2 2279 . . 3 (∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
1817opabbii 4715 . 2 {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19 dfoprab2 6698 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 dfoprab2 6698 . 2 {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2653 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wex 1703  wnf 1707  cop 4181  {copab 4710  {coprab 6648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-opab 4711  df-oprab 6651
This theorem is referenced by:  cbvoprab12v  6727  cbvmpt2x  6730  dfoprab4f  7223  fmpt2x  7233  tposoprab  7385  cbvmpt2x2  41885
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