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Theorem difab 3313
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 2102 . . 3  |-  ( y  e.  { x  |  ( ph  /\  -.  ps ) }  <->  [ y  /  x ] ( ph  /\ 
-.  ps ) )
2 sban 1904 . . 3  |-  ( [ y  /  x ]
( ph  /\  -.  ps ) 
<->  ( [ y  /  x ] ph  /\  [
y  /  x ]  -.  ps ) )
3 df-clab 2102 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
43bicomi 131 . . . 4  |-  ( [ y  /  x ] ph 
<->  y  e.  { x  |  ph } )
5 sbn 1901 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
6 df-clab 2102 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
75, 6xchbinxr 655 . . . 4  |-  ( [ y  /  x ]  -.  ps  <->  -.  y  e.  { x  |  ps }
)
84, 7anbi12i 453 . . 3  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ]  -.  ps )  <->  ( y  e. 
{ x  |  ph }  /\  -.  y  e. 
{ x  |  ps } ) )
91, 2, 83bitrri 206 . 2  |-  ( ( y  e.  { x  |  ph }  /\  -.  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
-.  ps ) } )
109difeqri 3164 1  |-  ( { x  |  ph }  \  { x  |  ps } )  =  {
x  |  ( ph  /\ 
-.  ps ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1314    e. wcel 1463   [wsb 1718   {cab 2101    \ cdif 3036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041
This theorem is referenced by:  notab  3314  difrab  3318  notrab  3321  imadiflem  5170  imadif  5171
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