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Theorem invdif 3173
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 2554 . . . . 5 x V
2 eldif 2921 . . . . 5 (x (V ∖ B) ↔ (x V ¬ x B))
31, 2mpbiran 846 . . . 4 (x (V ∖ B) ↔ ¬ x B)
43anbi2i 430 . . 3 ((x A x (V ∖ B)) ↔ (x A ¬ x B))
5 elin 3120 . . 3 (x (A ∩ (V ∖ B)) ↔ (x A x (V ∖ B)))
6 eldif 2921 . . 3 (x (AB) ↔ (x A ¬ x B))
74, 5, 63bitr4i 201 . 2 (x (A ∩ (V ∖ B)) ↔ x (AB))
87eqriv 2034 1 (A ∩ (V ∖ B)) = (AB)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cdif 2908  cin 2910
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918
This theorem is referenced by:  indif2  3175  difundir  3184  difindir  3186  difdif2ss  3188  difun1  3191  difdifdirss  3301  nn0supp  7990
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