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Theorem unopab 4651
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 3849 . . 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 1798 . . . . 5 (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3 andi 906 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)) ↔ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
43exbii 1763 . . . . . . 7 (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)) ↔ ∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
5 19.43 1798 . . . . . . 7 (∃𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
64, 5bitr2i 263 . . . . . 6 ((∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
76exbii 1763 . . . . 5 (∃𝑥(∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
82, 7bitr3i 264 . . . 4 ((∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓)))
98abbii 2722 . . 3 {𝑧 ∣ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ∨ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
101, 9eqtri 2628 . 2 ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}) = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
11 df-opab 4635 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
12 df-opab 4635 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
1311, 12uneq12i 3723 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = ({𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ∪ {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
14 df-opab 4635 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝜑𝜓))}
1510, 13, 143eqtr4i 2638 1 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wo 381  wa 382   = wceq 1474  wex 1694  {cab 2592  cun 3534  cop 4127  {copab 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-v 3171  df-un 3541  df-opab 4635
This theorem is referenced by:  xpundi  5081  xpundir  5082  cnvun  5440  coundi  5536  coundir  5537  mptun  5921  opsrtoslem1  19248  lgsquadlem3  24821
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