Theorem difdif 3559

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		pm4.45im 565	. . 3 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
2		iman 426	. . . . 5 |- ( ( x e. B -> x e. A ) <-> -. ( x e. B /\ -. x e. A ) )
3		eldif 3414	. . . . 5 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
4	2, 3	xchbinxr 313	. . . 4 |- ( ( x e. B -> x e. A ) <-> -. x e. ( B \ A ) )
5	4	anbi2i 700	. . 3 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) <-> ( x e. A /\ -. x e. ( B \ A ) ) )
6	1, 5	bitr2i 254	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
7	6	difeqri 3553	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   \ cdif 3401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-dif 3407

This theorem is referenced by:  dif0  3837  undifabs  3844