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Theorem rabxm 3794
Description: Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabxm  |-  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rabxm
StepHypRef Expression
1 rabid2 3021 . . 3  |-  ( A  =  { x  e.  A  |  ( ph  \/  -.  ph ) }  <->  A. x  e.  A  ( ph  \/  -.  ph ) )
2 exmidd 416 . . 3  |-  ( x  e.  A  ->  ( ph  \/  -.  ph )
)
31, 2mprgbir 2807 . 2  |-  A  =  { x  e.  A  |  ( ph  \/  -.  ph ) }
4 unrab 3754 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  \/  -.  ph ) }
53, 4eqtr4i 2475 1  |-  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1383    e. wcel 1804   {crab 2797    u. cun 3459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rab 2802  df-v 3097  df-un 3466
This theorem is referenced by:  usgrafilem1  24387  ddemeas  28185  ballotth  28453  mbfposadd  30037  jm2.22  30912  usgfislem1  32282  usgfisALTlem1  32286
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