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

Theorem comanbn 873
Description: Biconditional commutation law. (Contributed by NM, 1-Dec-1999.)
Assertion
Ref Expression
comanbn (ab ) C ((ac) ∩ (bc))

Proof of Theorem comanbn
StepHypRef Expression
1 comanb 872 . 2 (ab ) C ((ac ) ∩ (bc ))
2 conb 122 . . . 4 (ac) = (ac )
3 conb 122 . . . 4 (bc) = (bc )
42, 32an 79 . . 3 ((ac) ∩ (bc)) = ((ac ) ∩ (bc ))
54ax-r1 35 . 2 ((ac ) ∩ (bc )) = ((ac) ∩ (bc))
61, 5cbtr 182 1 (ab ) C ((ac) ∩ (bc))
Colors of variables: term
Syntax hints:   C wc 3   wn 4  tb 5  wa 7
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator