Theorem difdif 3097

Description: Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Assertion

Ref	Expression
difdif	|- ( A \ ( B \ A ) ) = A

Proof of Theorem difdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 107	. . 3 |- ( ( x e. A /\ -. x e. ( B \ A ) ) -> x e. A )
2		pm4.45im 327	. . . 4 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
3		imanim 818	. . . . . 6 |- ( ( x e. B -> x e. A ) -> -. ( x e. B /\ -. x e. A ) )
4		eldif 2982	. . . . . 6 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
5	3, 4	sylnibr 634	. . . . 5 |- ( ( x e. B -> x e. A ) -> -. x e. ( B \ A ) )
6	5	anim2i 334	. . . 4 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) -> ( x e. A /\ -. x e. ( B \ A ) ) )
7	2, 6	sylbi 119	. . 3 |- ( x e. A -> ( x e. A /\ -. x e. ( B \ A ) ) )
8	1, 7	impbii 124	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
9	8	difeqri 3092	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   \ cdif 2970

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975

This theorem is referenced by:  dif0  3314