Theorem difrab 3377

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 2441	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		df-rab 2441	. . 3 |- { x e. A | ps } = { x | ( x e. A /\ ps ) }
3	1, 2	difeq12i 3219	. 2 |- ( { x e. A | ph } \ { x e. A | ps } ) = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
4		df-rab 2441	. . 3 |- { x e. A | ( ph /\ -. ps ) } = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
5		difab 3372	. . . 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 2270	. . . 4 |- { x | ( x e. A /\ ( ph /\ -. ps ) ) } = { x | ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
17	5, 16	eqtr4i 2178	. . 3 |- ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } ) = { x | ( x e. A /\ ( ph /\ -. ps ) ) }
18	4, 17	eqtr4i 2178	. 2 |- { x e. A | ( ph /\ -. ps ) } = ( { x | ( x e. A /\ ph ) } \ { x | ( x e. A /\ ps ) } )
19	3, 18	eqtr4i 2178	1 |- ( { x e. A | ph } \ { x e. A | ps } ) = { x e. A | ( ph /\ -. ps ) }

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 2125   {cab 2140   {crab 2436   \ cdif 3095

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rab 2441  df-v 2711  df-dif 3100

This theorem is referenced by: (None)