Theorem difdif 3332

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 109	. . 3 |- ( ( x e. A /\ -. x e. ( B \ A ) ) -> x e. A )
2		pm4.45im 334	. . . 4 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
3		imanim 694	. . . . . 6 |- ( ( x e. B -> x e. A ) -> -. ( x e. B /\ -. x e. A ) )
4		eldif 3209	. . . . . 6 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
5	3, 4	sylnibr 683	. . . . 5 |- ( ( x e. B -> x e. A ) -> -. x e. ( B \ A ) )
6	5	anim2i 342	. . . 4 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) -> ( x e. A /\ -. x e. ( B \ A ) ) )
7	2, 6	sylbi 121	. . 3 |- ( x e. A -> ( x e. A /\ -. x e. ( B \ A ) ) )
8	1, 7	impbii 126	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
9	8	difeqri 3327	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   \ cdif 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202

This theorem is referenced by:  dif0  3565