Theorem difrab 3507

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 2537	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2537	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	difeq12i 3345	. 2 |- ( { x e. A | ph } \ { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
4		df-rab 2537	. . 3 |- { x e. A | ( ph /\ -. ps ) } = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
5		difab 3500	. . . 4 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
6		anass 405	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( x e. A /\ ( ph /\ -. ps ) ) )
7		simpr 110	. . . . . . . . 9 |- ( ( x e. A /\ ps ) -> ps )
8	7	con3i 641	. . . . . . . 8 |- ( -. ps -> -. ( x e. A /\ ps ) )
9	8	anim2i 342	. . . . . . 7 |- ( ( ( x e. A /\ ph ) /\ -. ps ) -> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
10		pm3.2 139	. . . . . . . . . 10 |- ( x e. A -> ( ps -> ( x e. A /\ ps ) ) )
11	10	adantr 276	. . . . . . . . 9 |- ( ( x e. A /\ ph ) -> ( ps -> ( x e. A /\ ps ) ) )
12	11	con3d 640	. . . . . . . 8 |- ( ( x e. A /\ ph ) -> ( -. ( x e. A /\ ps ) -> -. ps ) )
13	12	imdistani 449	. . . . . . 7 |- ( ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) -> ( ( x e. A /\ ph ) /\ -. ps ) )
14	9, 13	impbii 126	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
15	6, 14	bitr3i 186	. . . . 5 |- ( ( x e. A /\ ( ph /\ -. ps ) ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
16	15	abbii 2354	. . . 4 |- { x | ( x e. A /\ ( ph /\ -. ps ) ) } = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
17	5, 16	eqtr4i 2262	. . 3 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
18	4, 17	eqtr4i 2262	. 2 |- { x e. A | ( ph /\ -. ps ) } = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
19	3, 18	eqtr4i 2262	1 |- ( { x e. A | ph } \ { x e. A | ps } ) = { x e. A | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   \ cdif 3217

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222

This theorem is referenced by: (None)