Theorem difdif 3736

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 585	. . 3 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
2		iman 440	. . . . 5 |- ( ( x e. B -> x e. A ) <-> -. ( x e. B /\ -. x e. A ) )
3		eldif 3584	. . . . 5 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
4	2, 3	xchbinxr 325	. . . 4 |- ( ( x e. B -> x e. A ) <-> -. x e. ( B \ A ) )
5	4	anbi2i 730	. . 3 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) <-> ( x e. A /\ -. x e. ( B \ A ) ) )
6	1, 5	bitr2i 265	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
7	6	difeqri 3730	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577

This theorem is referenced by:  dif0  3950  undifabs  4045