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Theorem rabxm 3477
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 2717 . . 3  |-  ( A  =  { x  e.  A  |  ( ph  \/  -.  ph ) }  <->  A. x  e.  A  ( ph  \/  -.  ph ) )
2 exmid 404 . . . 4  |-  ( ph  \/  -.  ph )
32a1i 10 . . 3  |-  ( x  e.  A  ->  ( ph  \/  -.  ph )
)
41, 3mprgbir 2613 . 2  |-  A  =  { x  e.  A  |  ( ph  \/  -.  ph ) }
5 unrab 3439 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )  =  {
x  e.  A  | 
( ph  \/  -.  ph ) }
64, 5eqtr4i 2306 1  |-  A  =  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  -.  ph } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1623    e. wcel 1684   {crab 2547    u. cun 3150
This theorem is referenced by:  ballotth  23096  rabxmOLD  26352  jm2.22  27088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-un 3157
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