Theorem difdif 3591

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 564	. . 3 |- ( x e. A <-> ( x e. A /\ ( x e. B -> x e. A ) ) )
2		iman 425	. . . . 5 |- ( ( x e. B -> x e. A ) <-> -. ( x e. B /\ -. x e. A ) )
3		eldif 3446	. . . . 5 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
4	2, 3	xchbinxr 312	. . . 4 |- ( ( x e. B -> x e. A ) <-> -. x e. ( B \ A ) )
5	4	anbi2i 698	. . 3 |- ( ( x e. A /\ ( x e. B -> x e. A ) ) <-> ( x e. A /\ -. x e. ( B \ A ) ) )
6	1, 5	bitr2i 253	. 2 |- ( ( x e. A /\ -. x e. ( B \ A ) ) <-> x e. A )
7	6	difeqri 3585	1 |- ( A \ ( B \ A ) ) = A

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1868   \ cdif 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439

This theorem is referenced by:  dif0  3865  undifabs  3872