ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  indifdir Structured version   Unicode version

Theorem indifdir 3187
Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
Assertion
Ref Expression
indifdir  \  i^i  C  i^i  C  \  i^i  C

Proof of Theorem indifdir
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . 4  i^i  C  C
2 elin 3120 . . . . 5  i^i  C  C
32notbii 593 . . . 4  i^i  C  C
41, 3anbi12i 433 . . 3  i^i  C  i^i  C  C  C
5 eldif 2921 . . 3  i^i  C  \  i^i  C  i^i  C  i^i  C
6 elin 3120 . . . . 5 
\  i^i 
C  \  C
7 eldif 2921 . . . . . 6  \
87anbi1i 431 . . . . 5 
\  C  C
96, 8bitri 173 . . . 4 
\  i^i 
C  C
10 an32 496 . . . . 5  C  C
11 simpl 102 . . . . . . . 8  C
1211con3i 561 . . . . . . 7  C
1312anim2i 324 . . . . . 6  C  C  C
14 simpl 102 . . . . . . 7  C  C  C
15 ax-in2 545 . . . . . . . . . . 11  C  C
1615expcomd 1327 . . . . . . . . . 10  C  C
1716impcom 116 . . . . . . . . 9  C  C
18 dfnot 1261 . . . . . . . . 9
1917, 18sylibr 137 . . . . . . . 8  C  C
2019adantll 445 . . . . . . 7  C  C
2114, 20jca 290 . . . . . 6  C  C  C
2213, 21impbii 117 . . . . 5  C  C  C
2310, 22bitri 173 . . . 4  C  C  C
249, 23bitri 173 . . 3 
\  i^i 
C  C  C
254, 5, 243bitr4ri 202 . 2 
\  i^i 
C  i^i  C  \  i^i  C
2625eqriv 2034 1  \  i^i  C  i^i  C  \  i^i  C
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wceq 1242   wfal 1247   wcel 1390    \ cdif 2908    i^i 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-bndl 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: (None)
  Copyright terms: Public domain W3C validator