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Theorem cbvoprab2 6874
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 1994 . . . . . . 7 𝑤 𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧
2 cbvoprab2.1 . . . . . . 7 𝑤𝜑
31, 2nfan 1979 . . . . . 6 𝑤(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
43nfex 2317 . . . . 5 𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)
5 nfv 1994 . . . . . . 7 𝑦 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧
6 cbvoprab2.2 . . . . . . 7 𝑦𝜓
75, 6nfan 1979 . . . . . 6 𝑦(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
87nfex 2317 . . . . 5 𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)
9 opeq2 4538 . . . . . . . . 9 (𝑦 = 𝑤 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑤⟩)
109opeq1d 4543 . . . . . . . 8 (𝑦 = 𝑤 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩)
1110eqeq2d 2780 . . . . . . 7 (𝑦 = 𝑤 → (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩))
12 cbvoprab2.3 . . . . . . 7 (𝑦 = 𝑤 → (𝜑𝜓))
1311, 12anbi12d 608 . . . . . 6 (𝑦 = 𝑤 → ((𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
1413exbidv 2001 . . . . 5 (𝑦 = 𝑤 → (∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)))
154, 8, 14cbvex 2432 . . . 4 (∃𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
1615exbii 1923 . . 3 (∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓))
1716abbii 2887 . 2 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑥𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
18 df-oprab 6796 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
19 df-oprab 6796 . 2 {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑥𝑤𝑧(𝑣 = ⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∧ 𝜓)}
2017, 18, 193eqtr4i 2802 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wex 1851  wnf 1855  {cab 2756  cop 4320  {coprab 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-oprab 6796
This theorem is referenced by:  cbvmpt22  39792
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