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Theorem difin 3168
Description: Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
difin (A ∖ (AB)) = (AB)

Proof of Theorem difin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ax-in2 545 . . . . . . . 8 (¬ (x A x B) → ((x A x B) → ⊥ ))
21expd 245 . . . . . . 7 (¬ (x A x B) → (x A → (x B → ⊥ )))
3 dfnot 1261 . . . . . . 7 x B ↔ (x B → ⊥ ))
42, 3syl6ibr 151 . . . . . 6 (¬ (x A x B) → (x A → ¬ x B))
54com12 27 . . . . 5 (x A → (¬ (x A x B) → ¬ x B))
65imdistani 419 . . . 4 ((x A ¬ (x A x B)) → (x A ¬ x B))
7 simpr 103 . . . . . 6 ((x A x B) → x B)
87con3i 561 . . . . 5 x B → ¬ (x A x B))
98anim2i 324 . . . 4 ((x A ¬ x B) → (x A ¬ (x A x B)))
106, 9impbii 117 . . 3 ((x A ¬ (x A x B)) ↔ (x A ¬ x B))
11 eldif 2921 . . . 4 (x (A ∖ (AB)) ↔ (x A ¬ x (AB)))
12 elin 3120 . . . . . 6 (x (AB) ↔ (x A x B))
1312notbii 593 . . . . 5 x (AB) ↔ ¬ (x A x B))
1413anbi2i 430 . . . 4 ((x A ¬ x (AB)) ↔ (x A ¬ (x A x B)))
1511, 14bitri 173 . . 3 (x (A ∖ (AB)) ↔ (x A ¬ (x A x B)))
16 eldif 2921 . . 3 (x (AB) ↔ (x A ¬ x B))
1710, 15, 163bitr4i 201 . 2 (x (A ∖ (AB)) ↔ x (AB))
1817eqriv 2034 1 (A ∖ (AB)) = (AB)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242  wfal 1247   wcel 1390  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-fal 1248  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:  inssddif  3172  symdif1  3196  notrab  3208
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