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Theorem difab 3268
Description: Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difab  |-  ( { x  |  ph }  \  { x  |  ps } )  =  {
x  |  ( ph  /\ 
-.  ps ) }

Proof of Theorem difab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2075 . . 3  |-  ( y  e.  { x  |  ( ph  /\  -.  ps ) }  <->  [ y  /  x ] ( ph  /\ 
-.  ps ) )
2 sban 1877 . . 3  |-  ( [ y  /  x ]
( ph  /\  -.  ps ) 
<->  ( [ y  /  x ] ph  /\  [
y  /  x ]  -.  ps ) )
3 df-clab 2075 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
43bicomi 130 . . . 4  |-  ( [ y  /  x ] ph 
<->  y  e.  { x  |  ph } )
5 sbn 1874 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
6 df-clab 2075 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
75, 6xchbinxr 643 . . . 4  |-  ( [ y  /  x ]  -.  ps  <->  -.  y  e.  { x  |  ps }
)
84, 7anbi12i 448 . . 3  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ]  -.  ps )  <->  ( y  e. 
{ x  |  ph }  /\  -.  y  e. 
{ x  |  ps } ) )
91, 2, 83bitrri 205 . 2  |-  ( ( y  e.  { x  |  ph }  /\  -.  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
-.  ps ) } )
109difeqri 3120 1  |-  ( { x  |  ph }  \  { x  |  ps } )  =  {
x  |  ( ph  /\ 
-.  ps ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    = wceq 1289    e. wcel 1438   [wsb 1692   {cab 2074    \ cdif 2996
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-v 2621  df-dif 3001
This theorem is referenced by:  notab  3269  difrab  3273  notrab  3276  imadiflem  5093  imadif  5094
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