Theorem difrab 3396

Description: Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)

Assertion

Ref	Expression
difrab	|- ( { x e. A | ph } \ { x e. A | ps } ) = { x e. A | ( ph /\ -. ps ) }

Proof of Theorem difrab

Step	Hyp	Ref	Expression
1		df-rab 2453	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2453	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	difeq12i 3238	. 2 |- ( { x e. A | ph } \ { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
4		df-rab 2453	. . 3 |- { x e. A | ( ph /\ -. ps ) } = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
5		difab 3391	. . . 4 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
6		anass 399	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( x e. A /\ ( ph /\ -. ps ) ) )
7		simpr 109	. . . . . . . . 9 |- ( ( x e. A /\ ps ) -> ps )
8	7	con3i 622	. . . . . . . 8 |- ( -. ps -> -. ( x e. A /\ ps ) )
9	8	anim2i 340	. . . . . . 7 |- ( ( ( x e. A /\ ph ) /\ -. ps ) -> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
10		pm3.2 138	. . . . . . . . . 10 |- ( x e. A -> ( ps -> ( x e. A /\ ps ) ) )
11	10	adantr 274	. . . . . . . . 9 |- ( ( x e. A /\ ph ) -> ( ps -> ( x e. A /\ ps ) ) )
12	11	con3d 621	. . . . . . . 8 |- ( ( x e. A /\ ph ) -> ( -. ( x e. A /\ ps ) -> -. ps ) )
13	12	imdistani 442	. . . . . . 7 |- ( ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) -> ( ( x e. A /\ ph ) /\ -. ps ) )
14	9, 13	impbii 125	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
15	6, 14	bitr3i 185	. . . . 5 |- ( ( x e. A /\ ( ph /\ -. ps ) ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
16	15	abbii 2282	. . . 4 |- { x | ( x e. A /\ ( ph /\ -. ps ) ) } = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
17	5, 16	eqtr4i 2189	. . 3 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
18	4, 17	eqtr4i 2189	. 2 |- { x e. A | ( ph /\ -. ps ) } = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
19	3, 18	eqtr4i 2189	1 |- ( { x e. A | ph } \ { x e. A | ps } ) = { x e. A | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   {cab 2151   {crab 2448   \ 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-ral 2449  df-rab 2453  df-v 2728  df-dif 3118

This theorem is referenced by: (None)