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Theorem indifdir 3511
 Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir ((A B) ∩ C) = ((AC) (BC))

Proof of Theorem indifdir
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 pm3.24 852 . . . . . . . 8 ¬ (x C ¬ x C)
21intnan 880 . . . . . . 7 ¬ (x A (x C ¬ x C))
3 anass 630 . . . . . . 7 (((x A x C) ¬ x C) ↔ (x A (x C ¬ x C)))
42, 3mtbir 290 . . . . . 6 ¬ ((x A x C) ¬ x C)
54biorfi 396 . . . . 5 (((x A x C) ¬ x B) ↔ (((x A x C) ¬ x B) ((x A x C) ¬ x C)))
6 an32 773 . . . . 5 (((x A ¬ x B) x C) ↔ ((x A x C) ¬ x B))
7 andi 837 . . . . 5 (((x A x C) x B ¬ x C)) ↔ (((x A x C) ¬ x B) ((x A x C) ¬ x C)))
85, 6, 73bitr4i 268 . . . 4 (((x A ¬ x B) x C) ↔ ((x A x C) x B ¬ x C)))
9 ianor 474 . . . . 5 (¬ (x B x C) ↔ (¬ x B ¬ x C))
109anbi2i 675 . . . 4 (((x A x C) ¬ (x B x C)) ↔ ((x A x C) x B ¬ x C)))
118, 10bitr4i 243 . . 3 (((x A ¬ x B) x C) ↔ ((x A x C) ¬ (x B x C)))
12 elin 3219 . . . 4 (x ((A B) ∩ C) ↔ (x (A B) x C))
13 eldif 3221 . . . . 5 (x (A B) ↔ (x A ¬ x B))
1413anbi1i 676 . . . 4 ((x (A B) x C) ↔ ((x A ¬ x B) x C))
1512, 14bitri 240 . . 3 (x ((A B) ∩ C) ↔ ((x A ¬ x B) x C))
16 eldif 3221 . . . 4 (x ((AC) (BC)) ↔ (x (AC) ¬ x (BC)))
17 elin 3219 . . . . 5 (x (AC) ↔ (x A x C))
18 elin 3219 . . . . . 6 (x (BC) ↔ (x B x C))
1918notbii 287 . . . . 5 x (BC) ↔ ¬ (x B x C))
2017, 19anbi12i 678 . . . 4 ((x (AC) ¬ x (BC)) ↔ ((x A x C) ¬ (x B x C)))
2116, 20bitri 240 . . 3 (x ((AC) (BC)) ↔ ((x A x C) ¬ (x B x C)))
2211, 15, 213bitr4i 268 . 2 (x ((A B) ∩ C) ↔ x ((AC) (BC)))
2322eqriv 2350 1 ((A B) ∩ C) = ((AC) (BC))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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: (None)
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