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Theorem difab 3748
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 2415 . . 3  |-  ( y  e.  { x  |  ( ph  /\  -.  ps ) }  <->  [ y  /  x ] ( ph  /\ 
-.  ps ) )
2 sban 2194 . . 3  |-  ( [ y  /  x ]
( ph  /\  -.  ps ) 
<->  ( [ y  /  x ] ph  /\  [
y  /  x ]  -.  ps ) )
3 df-clab 2415 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
43bicomi 205 . . . 4  |-  ( [ y  /  x ] ph 
<->  y  e.  { x  |  ph } )
5 sbn 2186 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
6 df-clab 2415 . . . . 5  |-  ( y  e.  { x  |  ps }  <->  [ y  /  x ] ps )
75, 6xchbinxr 312 . . . 4  |-  ( [ y  /  x ]  -.  ps  <->  -.  y  e.  { x  |  ps }
)
84, 7anbi12i 701 . . 3  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ]  -.  ps )  <->  ( y  e. 
{ x  |  ph }  /\  -.  y  e. 
{ x  |  ps } ) )
91, 2, 83bitrri 275 . 2  |-  ( ( y  e.  { x  |  ph }  /\  -.  y  e.  { x  |  ps } )  <->  y  e.  { x  |  ( ph  /\ 
-.  ps ) } )
109difeqri 3591 1  |-  ( { x  |  ph }  \  { x  |  ps } )  =  {
x  |  ( ph  /\ 
-.  ps ) }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437   [wsb 1789    e. wcel 1870   {cab 2414    \ cdif 3439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-dif 3445
This theorem is referenced by:  notab  3749  difrab  3753  notrab  3756
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