Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  3oa3 GIF version

Theorem 3oa3 1025
 Description: 3-variable orthoarguesion law expressed with the 3OA identity abbreviation. (Contributed by NM, 27-May-2004.)
Assertion
Ref Expression
3oa3 ((a1 c) ∩ (acOA b)) ≤ (b1 c)

Proof of Theorem 3oa3
StepHypRef Expression
1 df-id3oa 57 . . 3 (acOA b) = (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))
21lan 77 . 2 ((a1 c) ∩ (acOA b)) = ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c))))
3 3oa2 1024 . 2 ((a1 c) ∩ (((a1 c) ∩ (b1 c)) ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
42, 3bltr 138 1 ((a1 c) ∩ (acOA b)) ≤ (b1 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   ≡ wid3oa 27 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-3oa 998 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-id3oa 57  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator