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Theorem difrab 3271
Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
difrab  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  -.  ps ) }

Proof of Theorem difrab
StepHypRef Expression
1 df-rab 2368 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2368 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2difeq12i 3114 . 2  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  \  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2368 . . 3  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }
5 difab 3266 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) }
6 anass 393 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  <->  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) )
7 simpr 108 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ps )  ->  ps )
87con3i 597 . . . . . . . 8  |-  ( -. 
ps  ->  -.  ( x  e.  A  /\  ps )
)
98anim2i 334 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  ->  ( ( x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) )
10 pm3.2 137 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( ps  ->  ( x  e.  A  /\  ps )
) )
1110adantr 270 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ph )  ->  ( ps  ->  ( x  e.  A  /\  ps ) ) )
1211con3d 596 . . . . . . . 8  |-  ( ( x  e.  A  /\  ph )  ->  ( -.  ( x  e.  A  /\  ps )  ->  -.  ps ) )
1312imdistani 434 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps ) )  -> 
( ( x  e.  A  /\  ph )  /\  -.  ps ) )
149, 13impbii 124 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  <->  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps ) ) )
156, 14bitr3i 184 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  /\  -.  ps ) )  <->  ( (
x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) )
1615abbii 2203 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps )
) }
175, 16eqtr4i 2111 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  -.  ps )
) }
184, 17eqtr4i 2111 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  ( {
x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )
193, 18eqtr4i 2111 1  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  -.  ps ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   {crab 2363    \ cdif 2994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-dif 2999
This theorem is referenced by: (None)
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