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Theorem cbvoprab1 6892
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 1992 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab1.1 . . . . . 6 𝑤𝜑
31, 2nfan 1977 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 2301 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
5 nfv 1992 . . . . . 6 𝑥 𝑣 = ⟨𝑤, 𝑦
6 cbvoprab1.2 . . . . . 6 𝑥𝜓
75, 6nfan 1977 . . . . 5 𝑥(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
87nfex 2301 . . . 4 𝑥𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)
9 opeq1 4553 . . . . . . 7 (𝑥 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
109eqeq2d 2770 . . . . . 6 (𝑥 = 𝑤 → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑤, 𝑦⟩))
11 cbvoprab1.3 . . . . . 6 (𝑥 = 𝑤 → (𝜑𝜓))
1210, 11anbi12d 749 . . . . 5 (𝑥 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
1312exbidv 1999 . . . 4 (𝑥 = 𝑤 → (∃𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)))
144, 8, 13cbvex 2417 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓))
1514opabbii 4869 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
16 dfoprab2 6866 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 dfoprab2 6866 . 2 {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑤𝑦(𝑣 = ⟨𝑤, 𝑦⟩ ∧ 𝜓)}
1815, 16, 173eqtr4i 2792 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wnf 1857  cop 4327  {copab 4864  {coprab 6814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-oprab 6817
This theorem is referenced by:  cbvmpt21  39777
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