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Theorem phiun 4614
 Description: The phi operation distributes over union. (Contributed by SF, 20-Feb-2015.)
Assertion
Ref Expression
phiun Phi (AB) = ( Phi A Phi B)

Proof of Theorem phiun
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexun 3443 . . 3 (y (AB)x = if(y Nn , (y +c 1c), y) ↔ (y A x = if(y Nn , (y +c 1c), y) y B x = if(y Nn , (y +c 1c), y)))
21abbii 2465 . 2 {x y (AB)x = if(y Nn , (y +c 1c), y)} = {x (y A x = if(y Nn , (y +c 1c), y) y B x = if(y Nn , (y +c 1c), y))}
3 df-phi 4565 . 2 Phi (AB) = {x y (AB)x = if(y Nn , (y +c 1c), y)}
4 df-phi 4565 . . . 4 Phi A = {x y A x = if(y Nn , (y +c 1c), y)}
5 df-phi 4565 . . . 4 Phi B = {x y B x = if(y Nn , (y +c 1c), y)}
64, 5uneq12i 3416 . . 3 ( Phi A Phi B) = ({x y A x = if(y Nn , (y +c 1c), y)} ∪ {x y B x = if(y Nn , (y +c 1c), y)})
7 unab 3521 . . 3 ({x y A x = if(y Nn , (y +c 1c), y)} ∪ {x y B x = if(y Nn , (y +c 1c), y)}) = {x (y A x = if(y Nn , (y +c 1c), y) y B x = if(y Nn , (y +c 1c), y))}
86, 7eqtri 2373 . 2 ( Phi A Phi B) = {x (y A x = if(y Nn , (y +c 1c), y) y B x = if(y Nn , (y +c 1c), y))}
92, 3, 83eqtr4i 2383 1 Phi (AB) = ( Phi A Phi B)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615   ∪ cun 3207   ifcif 3662  1cc1c 4134   Nn cnnc 4373   +c cplc 4375   Phi cphi 4562 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-phi 4565 This theorem is referenced by:  phialllem2  4617
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