QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  bi1o1a Structured version   Unicode version

Theorem bi1o1a 798
Description: Equivalence to biconditional.
Assertion
Ref Expression
bi1o1a (a == b) = ((a ->1 (a ^ b)) ^ ((a v b) ->1 a))

Proof of Theorem bi1o1a
StepHypRef Expression
1 lea 160 . . . . . . 7 (a' ^ b') =< a'
2 leo 158 . . . . . . 7 a' =< (a' v (a ^ b))
31, 2letr 137 . . . . . 6 (a' ^ b') =< (a' v (a ^ b))
43lecom 180 . . . . 5 (a' ^ b') C (a' v (a ^ b))
54comcom 453 . . . 4 (a' v (a ^ b)) C (a' ^ b')
6 comor1 461 . . . . 5 (a' v (a ^ b)) C a'
76comcom7 460 . . . 4 (a' v (a ^ b)) C a
85, 7fh1 469 . . 3 ((a' v (a ^ b)) ^ ((a' ^ b') v a)) = (((a' v (a ^ b)) ^ (a' ^ b')) v ((a' v (a ^ b)) ^ a))
98ax-r1 35 . 2 (((a' v (a ^ b)) ^ (a' ^ b')) v ((a' v (a ^ b)) ^ a)) = ((a' v (a ^ b)) ^ ((a' ^ b') v a))
10 dfb 94 . . 3 (a == b) = ((a ^ b) v (a' ^ b'))
11 ax-a2 31 . . 3 ((a ^ b) v (a' ^ b')) = ((a' ^ b') v (a ^ b))
12 leid 148 . . . . . 6 (a' ^ b') =< (a' ^ b')
133, 12ler2an 173 . . . . 5 (a' ^ b') =< ((a' v (a ^ b)) ^ (a' ^ b'))
14 lear 161 . . . . 5 ((a' v (a ^ b)) ^ (a' ^ b')) =< (a' ^ b')
1513, 14lebi 145 . . . 4 (a' ^ b') = ((a' v (a ^ b)) ^ (a' ^ b'))
16 dff 101 . . . . . . 7 0 = (a ^ a')
17 ancom 74 . . . . . . 7 (a ^ a') = (a' ^ a)
1816, 17ax-r2 36 . . . . . 6 0 = (a' ^ a)
1918ax-r5 38 . . . . 5 (0 v ((a ^ b) ^ a)) = ((a' ^ a) v ((a ^ b) ^ a))
20 lea 160 . . . . . . . 8 (a ^ b) =< a
2120df2le2 136 . . . . . . 7 ((a ^ b) ^ a) = (a ^ b)
2221ax-r1 35 . . . . . 6 (a ^ b) = ((a ^ b) ^ a)
23 or0r 103 . . . . . . 7 (0 v ((a ^ b) ^ a)) = ((a ^ b) ^ a)
2423ax-r1 35 . . . . . 6 ((a ^ b) ^ a) = (0 v ((a ^ b) ^ a))
2522, 24ax-r2 36 . . . . 5 (a ^ b) = (0 v ((a ^ b) ^ a))
26 comid 187 . . . . . . 7 a C a
2726comcom2 183 . . . . . 6 a C a'
28 comanr1 464 . . . . . 6 a C (a ^ b)
2927, 28fh1r 473 . . . . 5 ((a' v (a ^ b)) ^ a) = ((a' ^ a) v ((a ^ b) ^ a))
3019, 25, 293tr1 63 . . . 4 (a ^ b) = ((a' v (a ^ b)) ^ a)
3115, 302or 72 . . 3 ((a' ^ b') v (a ^ b)) = (((a' v (a ^ b)) ^ (a' ^ b')) v ((a' v (a ^ b)) ^ a))
3210, 11, 313tr 65 . 2 (a == b) = (((a' v (a ^ b)) ^ (a' ^ b')) v ((a' v (a ^ b)) ^ a))
33 df-i1 44 . . . 4 (a ->1 (a ^ b)) = (a' v (a ^ (a ^ b)))
34 lear 161 . . . . . 6 (a ^ (a ^ b)) =< (a ^ b)
35 leid 148 . . . . . . 7 (a ^ b) =< (a ^ b)
3620, 35ler2an 173 . . . . . 6 (a ^ b) =< (a ^ (a ^ b))
3734, 36lebi 145 . . . . 5 (a ^ (a ^ b)) = (a ^ b)
3837lor 70 . . . 4 (a' v (a ^ (a ^ b))) = (a' v (a ^ b))
3933, 38ax-r2 36 . . 3 (a ->1 (a ^ b)) = (a' v (a ^ b))
40 df-i1 44 . . . 4 ((a v b) ->1 a) = ((a v b)' v ((a v b) ^ a))
41 anor3 90 . . . . . 6 (a' ^ b') = (a v b)'
4241ax-r1 35 . . . . 5 (a v b)' = (a' ^ b')
43 lear 161 . . . . . 6 ((a v b) ^ a) =< a
44 leo 158 . . . . . . 7 a =< (a v b)
45 leid 148 . . . . . . 7 a =< a
4644, 45ler2an 173 . . . . . 6 a =< ((a v b) ^ a)
4743, 46lebi 145 . . . . 5 ((a v b) ^ a) = a
4842, 472or 72 . . . 4 ((a v b)' v ((a v b) ^ a)) = ((a' ^ b') v a)
4940, 48ax-r2 36 . . 3 ((a v b) ->1 a) = ((a' ^ b') v a)
5039, 492an 79 . 2 ((a ->1 (a ^ b)) ^ ((a v b) ->1 a)) = ((a' v (a ^ b)) ^ ((a' ^ b') v a))
519, 32, 503tr1 63 1 (a == b) = ((a ->1 (a ^ b)) ^ ((a v b) ->1 a))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  0wf 9   ->1 wi1 12
This theorem is referenced by:  mlaconj  845
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
  Copyright terms: Public domain W3C validator