Theorem difab 3391

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.

Step	Hyp	Ref	Expression
1		df-clab 2152	. . 3 |- ( y e. { x | ( ph /\ -. ps ) } <-> [ y / x ] ( ph /\ -. ps ) )
2		sban 1943	. . 3 |- ( [ y / x ] ( ph /\ -. ps ) <-> ( [ y / x ] ph /\ [ y / x ] -. ps ) )
3		df-clab 2152	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	3	bicomi 131	. . . 4 |- ( [ y / x ] ph <-> y e. { x | ph } )
5		sbn 1940	. . . . 5 |- ( [ y / x ] -. ps <-> -. [ y / x ] ps )
6		df-clab 2152	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
7	5, 6	xchbinxr 673	. . . 4 |- ( [ y / x ] -. ps <-> -. y e. { x | ps } )
8	4, 7	anbi12i 456	. . 3 |- ( ( [ y / x ] ph /\ [ y / x ] -. ps ) <-> ( y e. { x | ph } /\ -. y e. { x | ps } ) )
9	1, 2, 8	3bitrri 206	. 2 |- ( ( y e. { x | ph } /\ -. y e. { x | ps } ) <-> y e. { x | ( ph /\ -. ps ) } )
10	9	difeqri 3242	1 |- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   /\ wa 103   = wceq 1343   [wsb 1750   e. wcel 2136   {cab 2151   \ cdif 3113

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118

This theorem is referenced by:  notab  3392  difrab  3396  notrab  3399  imadiflem  5267  imadif  5268