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Metamath Proof Explorer

Theorem difrab

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

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

Proof

Step	Hyp	Ref	Expression
1		df-rab	\|- { x e. A \| ph } = { x \| ( x e. A /\ ph ) }
2		df-rab	\|- { x e. A \| ps } = { x \| ( x e. A /\ ps ) }
3	1 2	difeq12i	\|- ( { x e. A \| ph } \ { x e. A \| ps } ) = ( { x \| ( x e. A /\ ph ) } \ { x \| ( x e. A /\ ps ) } )
4		df-rab	\|- { x e. A \| ( ph /\ -. ps ) } = { x \| ( x e. A /\ ( ph /\ -. ps ) ) }
5		difab	\|- ( { x \| ( x e. A /\ ph ) } \ { x \| ( x e. A /\ ps ) } ) = { x \| ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
6		anass	\|- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( x e. A /\ ( ph /\ -. ps ) ) )
7		simpr	\|- ( ( x e. A /\ ps ) -> ps )
8	7	con3i	\|- ( -. ps -> -. ( x e. A /\ ps ) )
9	8	anim2i	\|- ( ( ( x e. A /\ ph ) /\ -. ps ) -> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
10		pm3.2	\|- ( x e. A -> ( ps -> ( x e. A /\ ps ) ) )
11	10	adantr	\|- ( ( x e. A /\ ph ) -> ( ps -> ( x e. A /\ ps ) ) )
12	11	con3d	\|- ( ( x e. A /\ ph ) -> ( -. ( x e. A /\ ps ) -> -. ps ) )
13	12	imdistani	\|- ( ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) -> ( ( x e. A /\ ph ) /\ -. ps ) )
14	9 13	impbii	\|- ( ( ( x e. A /\ ph ) /\ -. ps ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
15	6 14	bitr3i	\|- ( ( x e. A /\ ( ph /\ -. ps ) ) <-> ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) )
16	15	abbii	\|- { x \| ( x e. A /\ ( ph /\ -. ps ) ) } = { x \| ( ( x e. A /\ ph ) /\ -. ( x e. A /\ ps ) ) }
17	5 16	eqtr4i	\|- ( { x \| ( x e. A /\ ph ) } \ { x \| ( x e. A /\ ps ) } ) = { x \| ( x e. A /\ ( ph /\ -. ps ) ) }
18	4 17	eqtr4i	\|- { x e. A \| ( ph /\ -. ps ) } = ( { x \| ( x e. A /\ ph ) } \ { x \| ( x e. A /\ ps ) } )
19	3 18	eqtr4i	\|- ( { x e. A \| ph } \ { x e. A \| ps } ) = { x e. A \| ( ph /\ -. ps ) }