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Theorem cbvoprab1 7440
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 1917 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab1.1 . . . . . 6 𝑤𝜑
31, 2nfan 1902 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 2317 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5 nfv 1917 . . . . . 6 𝑥 𝑣 = ⟨𝑤, 𝑦
6 cbvoprab1.2 . . . . . 6 𝑥𝜓
75, 6nfan 1902 . . . . 5 𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
87nfex 2317 . . . 4 𝑥𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9 opeq1 4828 . . . . . . 7 (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
109eqeq2d 2747 . . . . . 6 (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11 cbvoprab1.3 . . . . . 6 (𝑥 = 𝑤 → (𝜑𝜓))
1210, 11anbi12d 631 . . . . 5 (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
1312exbidv 1924 . . . 4 (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
144, 8, 13cbvexv1 2338 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
1514opabbii 5170 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16 dfoprab2 7411 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 dfoprab2 7411 . 2 {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
1815, 16, 173eqtr4i 2774 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wnf 1785  cop 4590  {copab 5165  {coprab 7354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166  df-oprab 7357
This theorem is referenced by:  cbvmpo1  43250
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