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Theorem cbvoprab1 6769
Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1 𝑤𝜑
cbvoprab1.2 𝑥𝜓
cbvoprab1.3 (𝑥 = 𝑤 → (𝜑𝜓))
Assertion
Ref Expression
cbvoprab1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvoprab1
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1883 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab1.1 . . . . . 6 𝑤𝜑
31, 2nfan 1868 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 2192 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5 nfv 1883 . . . . . 6 𝑥 𝑣 = ⟨𝑤, 𝑦
6 cbvoprab1.2 . . . . . 6 𝑥𝜓
75, 6nfan 1868 . . . . 5 𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
87nfex 2192 . . . 4 𝑥𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9 opeq1 4433 . . . . . . 7 (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
109eqeq2d 2661 . . . . . 6 (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11 cbvoprab1.3 . . . . . 6 (𝑥 = 𝑤 → (𝜑𝜓))
1210, 11anbi12d 747 . . . . 5 (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
1312exbidv 1890 . . . 4 (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
144, 8, 13cbvex 2308 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
1514opabbii 4750 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16 dfoprab2 6743 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 dfoprab2 6743 . 2 {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
1815, 16, 173eqtr4i 2683 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 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:  cbvmpt21  39592
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