Theorem difrab 3245

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 2358	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2358	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	difeq12i 3089	. 2 |- ( { x e. A | ph } \ { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
4		df-rab 2358	. . 3 |- { x e. A | ( ph /\ -. ps ) } = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
5		difab 3240	. . . 4 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
6		anass 393	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( x e. A /\ ( ph /\ -. ps ) ) )
7		simpr 108	. . . . . . . . 9 |- ( ( x e. A /\ ps ) -> ps )
8	7	con3i 595	. . . . . . . 8 |- ( -. ps -> -. ( x e. A /\ ps ) )
9	8	anim2i 334	. . . . . . 7 |- ( ( ( x e. A /\ ph ) /\ -. ps ) -> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
10		pm3.2 137	. . . . . . . . . 10 |- ( x e. A -> ( ps -> ( x e. A /\ ps ) ) )
11	10	adantr 270	. . . . . . . . 9 |- ( ( x e. A /\ ph ) -> ( ps -> ( x e. A /\ ps ) ) )
12	11	con3d 594	. . . . . . . 8 |- ( ( x e. A /\ ph ) -> ( -. ( x e. A /\ ps ) -> -. ps ) )
13	12	imdistani 434	. . . . . . 7 |- ( ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) -> ( ( x e. A /\ ph ) /\ -. ps ) )
14	9, 13	impbii 124	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
15	6, 14	bitr3i 184	. . . . 5 |- ( ( x e. A /\ ( ph /\ -. ps ) ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
16	15	abbii 2195	. . . 4 |- { x | ( x e. A /\ ( ph /\ -. ps ) ) } = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
17	5, 16	eqtr4i 2105	. . 3 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
18	4, 17	eqtr4i 2105	. 2 |- { x e. A | ( ph /\ -. ps ) } = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
19	3, 18	eqtr4i 2105	1 |- ( { x e. A | ph } \ { x e. A | ps } ) = { x e. A | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   {cab 2068   {crab 2353   \ cdif 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976

This theorem is referenced by: (None)