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Theorem cbvoprab1 5758
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 1473 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab1.1 . . . . . 6 𝑤𝜑
31, 2nfan 1509 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 1580 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5 nfv 1473 . . . . . 6 𝑥 𝑣 = ⟨𝑤, 𝑦
6 cbvoprab1.2 . . . . . 6 𝑥𝜓
75, 6nfan 1509 . . . . 5 𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
87nfex 1580 . . . 4 𝑥𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9 opeq1 3644 . . . . . . 7 (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
109eqeq2d 2106 . . . . . 6 (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11 cbvoprab1.3 . . . . . 6 (𝑥 = 𝑤 → (𝜑𝜓))
1210, 11anbi12d 458 . . . . 5 (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
1312exbidv 1760 . . . 4 (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
144, 8, 13cbvex 1693 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
1514opabbii 3927 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16 dfoprab2 5734 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 dfoprab2 5734 . 2 {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
1815, 16, 173eqtr4i 2125 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wnf 1401  wex 1433  cop 3469  {copab 3920  {coprab 5691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-opab 3922  df-oprab 5694
This theorem is referenced by: (None)
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