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Theorem difrab 3769
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 2813 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2813 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2difeq12i 3606 . 2  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  \  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2813 . . 3  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }
5 difab 3764 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) }
6 anass 647 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  <->  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) )
7 simpr 459 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ps )  ->  ps )
87con3i 135 . . . . . . . 8  |-  ( -. 
ps  ->  -.  ( x  e.  A  /\  ps )
)
98anim2i 567 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  ->  ( ( x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) )
10 pm3.2 445 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( ps  ->  ( x  e.  A  /\  ps )
) )
1110adantr 463 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ph )  ->  ( ps  ->  ( x  e.  A  /\  ps ) ) )
1211con3d 133 . . . . . . . 8  |-  ( ( x  e.  A  /\  ph )  ->  ( -.  ( x  e.  A  /\  ps )  ->  -.  ps ) )
1312imdistani 688 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps ) )  -> 
( ( x  e.  A  /\  ph )  /\  -.  ps ) )
149, 13impbii 188 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  <->  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps ) ) )
156, 14bitr3i 251 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  /\  -.  ps ) )  <->  ( (
x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) )
1615abbii 2588 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps )
) }
175, 16eqtr4i 2486 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  -.  ps )
) }
184, 17eqtr4i 2486 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  ( {
x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )
193, 18eqtr4i 2486 1  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  /\  -.  ps ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   {crab 2808    \ cdif 3458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464
This theorem is referenced by:  alephsuc3  8946  shftmbl  22118  musum  23668  clwlknclwlkdifs  25165  aciunf1  27733
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