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Theorem difab 3767
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 2453 . . 3  |-  ( y  e.  { x  |  ( ph  /\  -.  ps ) }  <->  [ y  /  x ] ( ph  /\ 
-.  ps ) )
2 sban 2114 . . 3  |-  ( [ y  /  x ]
( ph  /\  -.  ps ) 
<->  ( [ y  /  x ] ph  /\  [
y  /  x ]  -.  ps ) )
3 df-clab 2453 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
43bicomi 202 . . . 4  |-  ( [ y  /  x ] ph 
<->  y  e.  { x  |  ph } )
5 sbn 2105 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
6 df-clab 2453 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
75, 6xchbinxr 311 . . . 4  |-  ( [ y  /  x ]  -.  ps  <->  -.  y  e.  { x  |  ps }
)
84, 7anbi12i 697 . . 3  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ]  -.  ps )  <->  ( y  e. 
{ x  |  ph }  /\  -.  y  e. 
{ x  |  ps } ) )
91, 2, 83bitrri 272 . 2  |-  ( ( y  e.  { x  |  ph }  /\  -.  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
-.  ps ) } )
109difeqri 3624 1  |-  ( { x  |  ph }  \  { x  |  ps } )  =  {
x  |  ( ph  /\ 
-.  ps ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379   [wsb 1711    e. wcel 1767   {cab 2452    \ cdif 3473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479
This theorem is referenced by:  notab  3768  difrab  3772  notrab  3775
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