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Theorem coundi 5082
 Description: Class composition distributes over union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
coundi (A (BC)) = ((A B) ∪ (A C))

Proof of Theorem coundi
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 4638 . . 3 ({x, y z(xBz zAy)} ∪ {x, y z(xCz zAy)}) = {x, y (z(xBz zAy) z(xCz zAy))}
2 brun 4692 . . . . . . . 8 (x(BC)z ↔ (xBz xCz))
32anbi1i 676 . . . . . . 7 ((x(BC)z zAy) ↔ ((xBz xCz) zAy))
4 andir 838 . . . . . . 7 (((xBz xCz) zAy) ↔ ((xBz zAy) (xCz zAy)))
53, 4bitri 240 . . . . . 6 ((x(BC)z zAy) ↔ ((xBz zAy) (xCz zAy)))
65exbii 1582 . . . . 5 (z(x(BC)z zAy) ↔ z((xBz zAy) (xCz zAy)))
7 19.43 1605 . . . . 5 (z((xBz zAy) (xCz zAy)) ↔ (z(xBz zAy) z(xCz zAy)))
86, 7bitr2i 241 . . . 4 ((z(xBz zAy) z(xCz zAy)) ↔ z(x(BC)z zAy))
98opabbii 4626 . . 3 {x, y (z(xBz zAy) z(xCz zAy))} = {x, y z(x(BC)z zAy)}
101, 9eqtri 2373 . 2 ({x, y z(xBz zAy)} ∪ {x, y z(xCz zAy)}) = {x, y z(x(BC)z zAy)}
11 df-co 4726 . . 3 (A B) = {x, y z(xBz zAy)}
12 df-co 4726 . . 3 (A C) = {x, y z(xCz zAy)}
1311, 12uneq12i 3416 . 2 ((A B) ∪ (A C)) = ({x, y z(xBz zAy)} ∪ {x, y z(xCz zAy)})
14 df-co 4726 . 2 (A (BC)) = {x, y z(x(BC)z zAy)}
1510, 13, 143eqtr4ri 2384 1 (A (BC)) = ((A B) ∪ (A C))
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∪ cun 3207  {copab 4622   class class class wbr 4639   ∘ ccom 4721 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-opab 4623  df-br 4640  df-co 4726 This theorem is referenced by: (None)
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