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Theorem difdif 3392
 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.
StepHypRef Expression
1 pm4.45im 545 . . 3 (x A ↔ (x A (x Bx A)))
2 iman 413 . . . . 5 ((x Bx A) ↔ ¬ (x B ¬ x A))
3 eldif 3221 . . . . 5 (x (B A) ↔ (x B ¬ x A))
42, 3xchbinxr 302 . . . 4 ((x Bx A) ↔ ¬ x (B A))
54anbi2i 675 . . 3 ((x A (x Bx A)) ↔ (x A ¬ x (B A)))
61, 5bitr2i 241 . 2 ((x A ¬ x (B A)) ↔ x A)
76difeqri 3387 1 (A (B A)) = A
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215 This theorem is referenced by:  dif0  3620  undifabs  3627
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