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Theorem cbvoprab12 6771
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 1883 . . . . 5 𝑤 𝑢 = ⟨𝑥, 𝑦
2 cbvoprab12.1 . . . . 5 𝑤𝜑
31, 2nfan 1868 . . . 4 𝑤(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1883 . . . . 5 𝑣 𝑢 = ⟨𝑥, 𝑦
5 cbvoprab12.2 . . . . 5 𝑣𝜑
64, 5nfan 1868 . . . 4 𝑣(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1883 . . . . 5 𝑥 𝑢 = ⟨𝑤, 𝑣
8 cbvoprab12.3 . . . . 5 𝑥𝜓
97, 8nfan 1868 . . . 4 𝑥(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
10 nfv 1883 . . . . 5 𝑦 𝑢 = ⟨𝑤, 𝑣
11 cbvoprab12.4 . . . . 5 𝑦𝜓
1210, 11nfan 1868 . . . 4 𝑦(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)
13 opeq12 4435 . . . . . 6 ((𝑥 = 𝑤𝑦 = 𝑣) → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑣⟩)
1413eqeq2d 2661 . . . . 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 2316 . . 3 (∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓))
1817opabbii 4750 . 2 {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
19 dfoprab2 6743 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑥𝑦(𝑢 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 dfoprab2 6743 . 2 {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑢, 𝑧⟩ ∣ ∃𝑤𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2683 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wnf 1748  cop 4216  {copab 4745  {coprab 6691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-oprab 6694
This theorem is referenced by:  cbvoprab12v  6772  cbvmpt2x  6775  dfoprab4f  7270  fmpt2x  7281  tposoprab  7433  f1od2  29627  cbvmpt2x2  42439
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