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Theorem i3abs3 524
Description: Antecedent absorption.
Assertion
Ref Expression
i3abs3 ((a ->3 b) ->3 ((a ->3 b) ->3 a)) = ((a ->3 b) ->3 a)
Proof of Theorem i3abs3
StepHypRef Expression
1 df-t 41 . . . . . . . 8 1 = (a v a')
21lan 77 . . . . . . 7 ((a ->3 b)' ^ 1) = ((a ->3 b)' ^ (a v a'))
3 an1 106 . . . . . . 7 ((a ->3 b)' ^ 1) = (a ->3 b)'
4 comi31 508 . . . . . . . . . 10 a C (a ->3 b)
54comcom 453 . . . . . . . . 9 (a ->3 b) C a
65comcom3 454 . . . . . . . 8 (a ->3 b)' C a
75comcom4 455 . . . . . . . 8 (a ->3 b)' C a'
86, 7fh1 469 . . . . . . 7 ((a ->3 b)' ^ (a v a')) = (((a ->3 b)' ^ a) v ((a ->3 b)' ^ a'))
92, 3, 83tr2 64 . . . . . 6 (a ->3 b)' = (((a ->3 b)' ^ a) v ((a ->3 b)' ^ a'))
109ax-r1 35 . . . . 5 (((a ->3 b)' ^ a) v ((a ->3 b)' ^ a')) = (a ->3 b)'
11 comid 187 . . . . . . . 8 (a ->3 b) C (a ->3 b)
1211comcom2 183 . . . . . . 7 (a ->3 b) C (a ->3 b)'
1312, 5fh1 469 . . . . . 6 ((a ->3 b) ^ ((a ->3 b)' v a)) = (((a ->3 b) ^ (a ->3 b)') v ((a ->3 b) ^ a))
14 ax-a2 31 . . . . . . 7 (0 v ((a ->3 b) ^ a)) = (((a ->3 b) ^ a) v 0)
15 dff 101 . . . . . . . 8 0 = ((a ->3 b) ^ (a ->3 b)')
1615ax-r5 38 . . . . . . 7 (0 v ((a ->3 b) ^ a)) = (((a ->3 b) ^ (a ->3 b)') v ((a ->3 b) ^ a))
17 or0 102 . . . . . . 7 (((a ->3 b) ^ a) v 0) = ((a ->3 b) ^ a)
1814, 16, 173tr2 64 . . . . . 6 (((a ->3 b) ^ (a ->3 b)') v ((a ->3 b) ^ a)) = ((a ->3 b) ^ a)
1913, 18ax-r2 36 . . . . 5 ((a ->3 b) ^ ((a ->3 b)' v a)) = ((a ->3 b) ^ a)
2010, 192or 72 . . . 4 ((((a ->3 b)' ^ a) v ((a ->3 b)' ^ a')) v ((a ->3 b) ^ ((a ->3 b)' v a))) = ((a ->3 b)' v ((a ->3 b) ^ a))
2112, 5fh4 472 . . . . 5 ((a ->3 b)' v ((a ->3 b) ^ a)) = (((a ->3 b)' v (a ->3 b)) ^ ((a ->3 b)' v a))
22 ax-a2 31 . . . . . . . . 9 ((a ->3 b)' v (a ->3 b)) = ((a ->3 b) v (a ->3 b)')
23 df-t 41 . . . . . . . . . 10 1 = ((a ->3 b) v (a ->3 b)')
2423ax-r1 35 . . . . . . . . 9 ((a ->3 b) v (a ->3 b)') = 1
2522, 24ax-r2 36 . . . . . . . 8 ((a ->3 b)' v (a ->3 b)) = 1
2625ran 78 . . . . . . 7 (((a ->3 b)' v (a ->3 b)) ^ ((a ->3 b)' v a)) = (1 ^ ((a ->3 b)' v a))
27 ancom 74 . . . . . . 7 (1 ^ ((a ->3 b)' v a)) = (((a ->3 b)' v a) ^ 1)
2826, 27ax-r2 36 . . . . . 6 (((a ->3 b)' v (a ->3 b)) ^ ((a ->3 b)' v a)) = (((a ->3 b)' v a) ^ 1)
29 an1 106 . . . . . 6 (((a ->3 b)' v a) ^ 1) = ((a ->3 b)' v a)
3028, 29ax-r2 36 . . . . 5 (((a ->3 b)' v (a ->3 b)) ^ ((a ->3 b)' v a)) = ((a ->3 b)' v a)
3121, 30ax-r2 36 . . . 4 ((a ->3 b)' v ((a ->3 b) ^ a)) = ((a ->3 b)' v a)
3220, 31ax-r2 36 . . 3 ((((a ->3 b)' ^ a) v ((a ->3 b)' ^ a')) v ((a ->3 b) ^ ((a ->3 b)' v a))) = ((a ->3 b)' v a)
3332ax-r1 35 . 2 ((a ->3 b)' v a) = ((((a ->3 b)' ^ a) v ((a ->3 b)' ^ a')) v ((a ->3 b) ^ ((a ->3 b)' v a)))
34 lem4 511 . 2 ((a ->3 b) ->3 ((a ->3 b) ->3 a)) = ((a ->3 b)' v a)
35 df-i3 46 . 2 ((a ->3 b) ->3 a) = ((((a ->3 b)' ^ a) v ((a ->3 b)' ^ a')) v ((a ->3 b) ^ ((a ->3 b)' v a)))
3633, 34, 353tr1 63 1 ((a ->3 b) ->3 ((a ->3 b) ->3 a)) = ((a ->3 b) ->3 a)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  i3th7  549  i3th8  550
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