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Theorem difrab 3245
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 2358 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2358 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2difeq12i 3089 . 2  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  \  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2358 . . 3  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }
5 difab 3240 . . . 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 595 . . . . . . . 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 594 . . . . . . . 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 2195 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps )
) }
175, 16eqtr4i 2105 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  -.  ps )
) }
184, 17eqtr4i 2105 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  ( {
x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )
193, 18eqtr4i 2105 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 1285    e. wcel 1434   {cab 2068   {crab 2353    \ cdif 2971
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976
This theorem is referenced by: (None)
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