QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u2lemc4 Unicode version
Theorem u2lemc4 702
Description: Lemma for Dishkant implication study.
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u2lemc4 (a ->2 b) = (a' v b)
Proof of Theorem u2lemc4
StepHypRef Expression
1 df-i2 45 . 2 (a ->2 b) = (b v (a' ^ b'))
2 ulemc3.1 . . . . 5 a C b
32comcom3 454 . . . 4 a' C b
42comcom4 455 . . . 4 a' C b'
53, 4fh4 472 . . 3 (b v (a' ^ b')) = ((b v a') ^ (b v b'))
6 ax-a2 31 . . . . 5 (b v a') = (a' v b)
7 df-t 41 . . . . . 6 1 = (b v b')
87ax-r1 35 . . . . 5 (b v b') = 1
96, 82an 79 . . . 4 ((b v a') ^ (b v b')) = ((a' v b) ^ 1)
10 an1 106 . . . 4 ((a' v b) ^ 1) = (a' v b)
119, 10ax-r2 36 . . 3 ((b v a') ^ (b v b')) = (a' v b)
125, 11ax-r2 36 . 2 (b v (a' ^ b')) = (a' v b)
131, 12ax-r2 36 1 (a ->2 b) = (a' v b)
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7  1wt 8   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u2lemle1  711
  Copyright terms: Public domain W3C validator