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Theorem unopab 5230
Description: Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
unopab ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}

Proof of Theorem unopab
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2739 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
21anbi1d 631 . . . . 5 (𝑧 = 𝑤 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
322exbidv 1922 . . . 4 (𝑧 = 𝑤 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
41anbi1d 631 . . . . 5 (𝑧 = 𝑤 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
542exbidv 1922 . . . 4 (𝑧 = 𝑤 → (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
63, 5unabw 4313 . . 3 ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑤 ∣ (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))}
7 19.43 1880 . . . . 5 (∃𝑥(∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
8 andi 1009 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)) ↔ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
98exbii 1845 . . . . . . 7 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)) ↔ ∃𝑦((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
10 19.43 1880 . . . . . . 7 (∃𝑦((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
119, 10bitr2i 276 . . . . . 6 ((∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
1211exbii 1845 . . . . 5 (∃𝑥(∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
137, 12bitr3i 277 . . . 4 ((∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
1413abbii 2807 . . 3 {𝑤 ∣ (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
156, 14eqtri 2763 . 2 ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
16 df-opab 5211 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
17 df-opab 5211 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1816, 17uneq12i 4176 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
19 df-opab 5211 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
2015, 18, 193eqtr4i 2773 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 847   = wceq 1537  wex 1776  {cab 2712  cun 3961  cop 4637  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-opab 5211
This theorem is referenced by:  xpundi  5757  xpundir  5758  cnvun  6165  coundi  6269  coundir  6270  mptun  6715  opsrtoslem1  22097  lgsquadlem3  27441
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