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Theorem invdif 3156
Description: Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
invdif (A ∩ (V ∖ B)) = (AB)

Proof of Theorem invdif
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . . . 5 x V
2 eldif 2904 . . . . 5 (x (V ∖ B) ↔ (x V ¬ x B))
31, 2mpbiran 835 . . . 4 (x (V ∖ B) ↔ ¬ x B)
43anbi2i 433 . . 3 ((x A x (V ∖ B)) ↔ (x A ¬ x B))
5 elin 3103 . . 3 (x (A ∩ (V ∖ B)) ↔ (x A x (V ∖ B)))
6 eldif 2904 . . 3 (x (AB) ↔ (x A ¬ x B))
74, 5, 63bitr4i 201 . 2 (x (A ∩ (V ∖ B)) ↔ x (AB))
87eqriv 2019 1 (A ∩ (V ∖ B)) = (AB)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1228   wcel 1374  Vcvv 2535  cdif 2891  cin 2893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-in 2901
This theorem is referenced by:  indif2  3158  difundir  3167  difindir  3169  difdif2ss  3171  difun1  3174  difdifdirss  3284
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