Theorem difab 3450

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 2194	. . 3 |- ( y e. { x | ( ph /\ -. ps ) } <-> [ y / x ] ( ph /\ -. ps ) )
2		sban 1984	. . 3 |- ( [ y / x ] ( ph /\ -. ps ) <-> ( [ y / x ] ph /\ [ y / x ] -. ps ) )
3		df-clab 2194	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
4	3	bicomi 132	. . . 4 |- ( [ y / x ] ph <-> y e. { x | ph } )
5		sbn 1981	. . . . 5 |- ( [ y / x ] -. ps <-> -. [ y / x ] ps )
6		df-clab 2194	. . . . 5 |- ( y e. { x | ps } <-> [ y / x ] ps )
7	5, 6	xchbinxr 685	. . . 4 |- ( [ y / x ] -. ps <-> -. y e. { x | ps } )
8	4, 7	anbi12i 460	. . 3 |- ( ( [ y / x ] ph /\ [ y / x ] -. ps ) <-> ( y e. { x | ph } /\ -. y e. { x | ps } ) )
9	1, 2, 8	3bitrri 207	. 2 |- ( ( y e. { x | ph } /\ -. y e. { x | ps } ) <-> y e. { x | ( ph /\ -. ps ) } )
10	9	difeqri 3301	1 |- ( { x | ph } \ { x | ps } ) = { x | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   /\ wa 104   = wceq 1373   [wsb 1786   e. wcel 2178   {cab 2193   \ cdif 3171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176

This theorem is referenced by:  notab  3451  difrab  3455  notrab  3458  imadiflem  5372  imadif  5373