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Theorem difin 3147
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 533 . . . . . . . 8 (¬ (x A x B) → ((x A x B) → ⊥ ))
21expd 245 . . . . . . 7 (¬ (x A x B) → (x A → (x B → ⊥ )))
3 dfnot 1245 . . . . . . 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 422 . . . 4 ((x A ¬ (x A x B)) → (x A ¬ x B))
7 ax-ia2 100 . . . . . 6 ((x A x B) → x B)
87con3i 549 . . . . 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 2900 . . . 4 (x (A ∖ (AB)) ↔ (x A ¬ x (AB)))
12 elin 3099 . . . . . 6 (x (AB) ↔ (x A x B))
1312notbii 581 . . . . 5 x (AB) ↔ ¬ (x A x B))
1413anbi2i 433 . . . 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 2900 . . 3 (x (AB) ↔ (x A ¬ x B))
1710, 15, 163bitr4i 201 . 2 (x (A ∖ (AB)) ↔ x (AB))
1817eqriv 2015 1 (A ∖ (AB)) = (AB)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1226  wfal 1231   wcel 1370  cdif 2887  cin 2889
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-dif 2893  df-in 2897
This theorem is referenced by:  inssddif  3151  symdif1  3175  notrab  3187
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