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Theorem unopab 4719
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 variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 unab 3886 . . 3 ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑧 ∣ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))}
2 19.43 1808 . . . . 5 (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3 andi 910 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)) ↔ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
43exbii 1772 . . . . . . 7 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)) ↔ ∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
5 19.43 1808 . . . . . . 7 (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
64, 5bitr2i 265 . . . . . 6 ((∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
76exbii 1772 . . . . 5 (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
82, 7bitr3i 266 . . . 4 ((∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
98abbii 2737 . . 3 {𝑧 ∣ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
101, 9eqtri 2642 . 2 ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
11 df-opab 4704 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
12 df-opab 4704 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1311, 12uneq12i 3757 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
14 df-opab 4704 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
1510, 13, 143eqtr4i 2652 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wo 383  wa 384   = wceq 1481  wex 1702  {cab 2606  cun 3565  cop 4174  {copab 4703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572  df-opab 4704
This theorem is referenced by:  xpundi  5161  xpundir  5162  cnvun  5526  coundi  5624  coundir  5625  mptun  6012  opsrtoslem1  19465  lgsquadlem3  25088
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