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Theorem difrab 3560
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 2660 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2660 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2difeq12i 3408 . 2  |-  ( { x  e.  A  |  ph }  \  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  \  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2660 . . 3  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }
5 difab 3555 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) }
6 anass 631 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  <->  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) )
7 simpr 448 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ps )  ->  ps )
87con3i 129 . . . . . . . 8  |-  ( -. 
ps  ->  -.  ( x  e.  A  /\  ps )
)
98anim2i 553 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  ->  ( ( x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) )
10 pm3.2 435 . . . . . . . . . 10  |-  ( x  e.  A  ->  ( ps  ->  ( x  e.  A  /\  ps )
) )
1110adantr 452 . . . . . . . . 9  |-  ( ( x  e.  A  /\  ph )  ->  ( ps  ->  ( x  e.  A  /\  ps ) ) )
1211con3d 127 . . . . . . . 8  |-  ( ( x  e.  A  /\  ph )  ->  ( -.  ( x  e.  A  /\  ps )  ->  -.  ps ) )
1312imdistani 672 . . . . . . 7  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps ) )  -> 
( ( x  e.  A  /\  ph )  /\  -.  ps ) )
149, 13impbii 181 . . . . . 6  |-  ( ( ( x  e.  A  /\  ph )  /\  -.  ps )  <->  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps ) ) )
156, 14bitr3i 243 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  /\  -.  ps ) )  <->  ( (
x  e.  A  /\  ph )  /\  -.  (
x  e.  A  /\  ps ) ) )
1615abbii 2501 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  /\ 
-.  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  /\  -.  ( x  e.  A  /\  ps )
) }
175, 16eqtr4i 2412 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  /\  -.  ps )
) }
184, 17eqtr4i 2412 . 2  |-  { x  e.  A  |  ( ph  /\  -.  ps ) }  =  ( {
x  |  ( x  e.  A  /\  ph ) }  \  { x  |  ( x  e.  A  /\  ps ) } )
193, 18eqtr4i 2412 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 359    = wceq 1649    e. wcel 1717   {cab 2375   {crab 2655    \ cdif 3262
This theorem is referenced by:  alephsuc3  8390  shftmbl  19302  musum  20845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rab 2660  df-v 2903  df-dif 3268
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