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Theorem cbvoprab3 6696
Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvoprab3.1 𝑤𝜑
cbvoprab3.2 𝑧𝜓
cbvoprab3.3 (𝑧 = 𝑤 → (𝜑𝜓))
Assertion
Ref Expression
cbvoprab3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvoprab3
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . . . . 6 𝑤 𝑣 = ⟨𝑥, 𝑦
2 cbvoprab3.1 . . . . . 6 𝑤𝜑
31, 2nfan 1825 . . . . 5 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
43nfex 2151 . . . 4 𝑤𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
54nfex 2151 . . 3 𝑤𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
6 nfv 1840 . . . . . 6 𝑧 𝑣 = ⟨𝑥, 𝑦
7 cbvoprab3.2 . . . . . 6 𝑧𝜓
86, 7nfan 1825 . . . . 5 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
98nfex 2151 . . . 4 𝑧𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
109nfex 2151 . . 3 𝑧𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
11 cbvoprab3.3 . . . . 5 (𝑧 = 𝑤 → (𝜑𝜓))
1211anbi2d 739 . . . 4 (𝑧 = 𝑤 → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
13122exbidv 1849 . . 3 (𝑧 = 𝑤 → (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
145, 10, 13cbvopab2 4696 . 2 {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
15 dfoprab2 6666 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑣, 𝑧⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
16 dfoprab2 6666 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} = {⟨𝑣, 𝑤⟩ ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1714, 15, 163eqtr4i 2653 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wnf 1705  cop 4161  {copab 4682  {coprab 6616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-opab 4684  df-oprab 6619
This theorem is referenced by:  cbvoprab3v  6697  tposoprab  7348  erovlem  7803
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