MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indifdir Structured version   Visualization version   GIF version

Theorem indifdir 3859
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Proof of Theorem indifdir
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pm3.24 925 . . . . . . . 8 ¬ (𝑥𝐶 ∧ ¬ 𝑥𝐶)
21intnan 959 . . . . . . 7 ¬ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐶))
3 anass 680 . . . . . . 7 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐶 ∧ ¬ 𝑥𝐶)))
42, 3mtbir 313 . . . . . 6 ¬ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)
54biorfi 422 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ↔ (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)))
6 an32 838 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵))
7 andi 910 . . . . 5 (((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)) ↔ (((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐵) ∨ ((𝑥𝐴𝑥𝐶) ∧ ¬ 𝑥𝐶)))
85, 6, 73bitr4i 292 . . . 4 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
9 ianor 509 . . . . 5 (¬ (𝑥𝐵𝑥𝐶) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶))
109anbi2i 729 . . . 4 (((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝑥𝐶)))
118, 10bitr4i 267 . . 3 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
12 elin 3774 . . . 4 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶))
13 eldif 3565 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
1413anbi1i 730 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
1512, 14bitri 264 . . 3 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝑥𝐶))
16 eldif 3565 . . . 4 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ (𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)))
17 elin 3774 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
18 elin 3774 . . . . . 6 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
1918notbii 310 . . . . 5 𝑥 ∈ (𝐵𝐶) ↔ ¬ (𝑥𝐵𝑥𝐶))
2017, 19anbi12i 732 . . . 4 ((𝑥 ∈ (𝐴𝐶) ∧ ¬ 𝑥 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2116, 20bitri 264 . . 3 (𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)) ↔ ((𝑥𝐴𝑥𝐶) ∧ ¬ (𝑥𝐵𝑥𝐶)))
2211, 15, 213bitr4i 292 . 2 (𝑥 ∈ ((𝐴𝐵) ∩ 𝐶) ↔ 𝑥 ∈ ((𝐴𝐶) ∖ (𝐵𝐶)))
2322eqriv 2618 1 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 383  wa 384   = wceq 1480  wcel 1987  cdif 3552  cin 3554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-in 3562
This theorem is referenced by:  preddif  5664  fresaun  6032  uniioombllem4  23260  subsalsal  39881
  Copyright terms: Public domain W3C validator