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Theorem difin 3492
 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)) = (A B)

Proof of Theorem difin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pm4.61 415 . . 3 (¬ (x Ax B) ↔ (x A ¬ x B))
2 anclb 530 . . . . 5 ((x Ax B) ↔ (x A → (x A x B)))
3 elin 3219 . . . . . 6 (x (AB) ↔ (x A x B))
43imbi2i 303 . . . . 5 ((x Ax (AB)) ↔ (x A → (x A x B)))
5 iman 413 . . . . 5 ((x Ax (AB)) ↔ ¬ (x A ¬ x (AB)))
62, 4, 53bitr2i 264 . . . 4 ((x Ax B) ↔ ¬ (x A ¬ x (AB)))
76con2bii 322 . . 3 ((x A ¬ x (AB)) ↔ ¬ (x Ax B))
8 eldif 3221 . . 3 (x (A B) ↔ (x A ¬ x B))
91, 7, 83bitr4i 268 . 2 ((x A ¬ x (AB)) ↔ x (A B))
109difeqri 3387 1 (A (AB)) = (A B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem is referenced by:  dfin4  3495  indif  3497  symdif1  3519  notrab  3532
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