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Theorem coundir 5083
 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
coundir ((AB) C) = ((A C) ∪ (B C))

Proof of Theorem coundir
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 4638 . . 3 ({x, z y(xCy yAz)} ∪ {x, z y(xCy yBz)}) = {x, z (y(xCy yAz) y(xCy yBz))}
2 brun 4692 . . . . . . . 8 (y(AB)z ↔ (yAz yBz))
32anbi2i 675 . . . . . . 7 ((xCy y(AB)z) ↔ (xCy (yAz yBz)))
4 andi 837 . . . . . . 7 ((xCy (yAz yBz)) ↔ ((xCy yAz) (xCy yBz)))
53, 4bitri 240 . . . . . 6 ((xCy y(AB)z) ↔ ((xCy yAz) (xCy yBz)))
65exbii 1582 . . . . 5 (y(xCy y(AB)z) ↔ y((xCy yAz) (xCy yBz)))
7 19.43 1605 . . . . 5 (y((xCy yAz) (xCy yBz)) ↔ (y(xCy yAz) y(xCy yBz)))
86, 7bitr2i 241 . . . 4 ((y(xCy yAz) y(xCy yBz)) ↔ y(xCy y(AB)z))
98opabbii 4626 . . 3 {x, z (y(xCy yAz) y(xCy yBz))} = {x, z y(xCy y(AB)z)}
101, 9eqtri 2373 . 2 ({x, z y(xCy yAz)} ∪ {x, z y(xCy yBz)}) = {x, z y(xCy y(AB)z)}
11 df-co 4726 . . 3 (A C) = {x, z y(xCy yAz)}
12 df-co 4726 . . 3 (B C) = {x, z y(xCy yBz)}
1311, 12uneq12i 3416 . 2 ((A C) ∪ (B C)) = ({x, z y(xCy yAz)} ∪ {x, z y(xCy yBz)})
14 df-co 4726 . 2 ((AB) C) = {x, z y(xCy y(AB)z)}
1510, 13, 143eqtr4ri 2384 1 ((AB) C) = ((A C) ∪ (B 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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