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Theorem cbvopab 4653
Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1 𝑧𝜑
cbvopab.2 𝑤𝜑
cbvopab.3 𝑥𝜓
cbvopab.4 𝑦𝜓
cbvopab.5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
Assertion
Ref Expression
cbvopab {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvopab
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 nfv 1830 . . . . 5 𝑧 𝑣 = ⟨𝑥, 𝑦
2 cbvopab.1 . . . . 5 𝑧𝜑
31, 2nfan 1816 . . . 4 𝑧(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
4 nfv 1830 . . . . 5 𝑤 𝑣 = ⟨𝑥, 𝑦
5 cbvopab.2 . . . . 5 𝑤𝜑
64, 5nfan 1816 . . . 4 𝑤(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
7 nfv 1830 . . . . 5 𝑥 𝑣 = ⟨𝑧, 𝑤
8 cbvopab.3 . . . . 5 𝑥𝜓
97, 8nfan 1816 . . . 4 𝑥(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
10 nfv 1830 . . . . 5 𝑦 𝑣 = ⟨𝑧, 𝑤
11 cbvopab.4 . . . . 5 𝑦𝜓
1210, 11nfan 1816 . . . 4 𝑦(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)
13 opeq12 4342 . . . . . 6 ((𝑥 = 𝑧𝑦 = 𝑤) → ⟨𝑥, 𝑦⟩ = ⟨𝑧, 𝑤⟩)
1413eqeq2d 2620 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = ⟨𝑥, 𝑦⟩ ↔ 𝑣 = ⟨𝑧, 𝑤⟩))
15 cbvopab.5 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))
1614, 15anbi12d 743 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)))
173, 6, 9, 12, 16cbvex2 2268 . . 3 (∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓))
1817abbii 2726 . 2 {𝑣 ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑣 ∣ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
19 df-opab 4644 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦(𝑣 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
20 df-opab 4644 . 2 {⟨𝑧, 𝑤⟩ ∣ 𝜓} = {𝑣 ∣ ∃𝑧𝑤(𝑣 = ⟨𝑧, 𝑤⟩ ∧ 𝜓)}
2118, 19, 203eqtr4i 2642 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  wnf 1699  {cab 2596  cop 4131  {copab 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644
This theorem is referenced by:  cbvopabv  4654  dfrel4  5504  aomclem8  36649
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