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Theorem brdif 4694
 Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011.)
Assertion
Ref Expression
brdif (A(R S)B ↔ (ARB ¬ ASB))

Proof of Theorem brdif
StepHypRef Expression
1 eldif 3221 . 2 (A, B (R S) ↔ (A, B R ¬ A, B S))
2 df-br 4640 . 2 (A(R S)BA, B (R S))
3 df-br 4640 . . 3 (ARBA, B R)
4 df-br 4640 . . . 4 (ASBA, B S)
54notbii 287 . . 3 ASB ↔ ¬ A, B S)
63, 5anbi12i 678 . 2 ((ARB ¬ ASB) ↔ (A, B R ¬ A, B S))
71, 2, 63bitr4i 268 1 (A(R S)B ↔ (ARB ¬ ASB))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   ∖ cdif 3206  ⟨cop 4561   class class class wbr 4639 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  df-br 4640 This theorem is referenced by:  fnfullfunlem1  5856  brltc  6114
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