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Theorem cbvoprab2 7363
Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab2.1 𝑤𝜑
cbvoprab2.2 𝑦𝜓
cbvoprab2.3 (𝑦 = 𝑤 → (𝜑𝜓))
Assertion
Ref Expression
cbvoprab2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑤,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvoprab2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1917 . . . . . . 7 𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧
2 cbvoprab2.1 . . . . . . 7 𝑤𝜑
31, 2nfan 1902 . . . . . 6 𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
43nfex 2318 . . . . 5 𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5 nfv 1917 . . . . . . 7 𝑦 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧
6 cbvoprab2.2 . . . . . . 7 𝑦𝜓
75, 6nfan 1902 . . . . . 6 𝑦(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
87nfex 2318 . . . . 5 𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
9 opeq2 4805 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
109opeq1d 4810 . . . . . . . 8 (𝑦 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
1110eqeq2d 2749 . . . . . . 7 (𝑦 = 𝑤 → (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
12 cbvoprab2.3 . . . . . . 7 (𝑦 = 𝑤 → (𝜑𝜓))
1311, 12anbi12d 631 . . . . . 6 (𝑦 = 𝑤 → ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
1413exbidv 1924 . . . . 5 (𝑦 = 𝑤 → (∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
154, 8, 14cbvexv1 2339 . . . 4 (∃𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
1615exbii 1850 . . 3 (∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
1716abbii 2808 . 2 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑥𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
18 df-oprab 7279 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19 df-oprab 7279 . 2 {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑥𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
2017, 18, 193eqtr4i 2776 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wnf 1786  {cab 2715  cop 4567  {coprab 7276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-oprab 7279
This theorem is referenced by:  cbvmpo2  42647
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