Theorem difdif 3206

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 108	. . 3 |- ( ( x e. A /\ -. x e. ( B \ A ) ) -> x e. A )
2		pm4.45im 332	. . . 4 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
3		imanim 678	. . . . . 6 |- ( ( x e. B -> x e. A ) -> -. ( x e. B /\ -. x e. A ) )
4		eldif 3085	. . . . . 6 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
5	3, 4	sylnibr 667	. . . . 5 |- ( ( x e. B -> x e. A ) -> -. x e. ( B \ A ) )
6	5	anim2i 340	. . . 4 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) -> ( x e. A /\ -. x e. ( B \ A ) ) )
7	2, 6	sylbi 120	. . 3 |- ( x e. A -> ( x e. A /\ -. x e. ( B \ A ) ) )
8	1, 7	impbii 125	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
9	8	difeqri 3201	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   \ cdif 3073

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078

This theorem is referenced by:  dif0  3438