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Theorem i3id 251
Description: Identity for Kalmbach implication.
Assertion
Ref Expression
i3id (a ->3 a) = 1
Proof of Theorem i3id
StepHypRef Expression
1 ancom 74 . . . . . . . 8 (a' ^ a) = (a ^ a')
2 dff 101 . . . . . . . . 9 0 = (a ^ a')
32ax-r1 35 . . . . . . . 8 (a ^ a') = 0
41, 3ax-r2 36 . . . . . . 7 (a' ^ a) = 0
5 anidm 111 . . . . . . 7 (a' ^ a') = a'
64, 52or 72 . . . . . 6 ((a' ^ a) v (a' ^ a')) = (0 v a')
7 ax-a2 31 . . . . . 6 (0 v a') = (a' v 0)
86, 7ax-r2 36 . . . . 5 ((a' ^ a) v (a' ^ a')) = (a' v 0)
9 or0 102 . . . . 5 (a' v 0) = a'
108, 9ax-r2 36 . . . 4 ((a' ^ a) v (a' ^ a')) = a'
11 ax-a2 31 . . . . . . 7 (a' v a) = (a v a')
12 df-t 41 . . . . . . . 8 1 = (a v a')
1312ax-r1 35 . . . . . . 7 (a v a') = 1
1411, 13ax-r2 36 . . . . . 6 (a' v a) = 1
1514lan 77 . . . . 5 (a ^ (a' v a)) = (a ^ 1)
16 an1 106 . . . . 5 (a ^ 1) = a
1715, 16ax-r2 36 . . . 4 (a ^ (a' v a)) = a
1810, 172or 72 . . 3 (((a' ^ a) v (a' ^ a')) v (a ^ (a' v a))) = (a' v a)
1918, 11ax-r2 36 . 2 (((a' ^ a) v (a' ^ a')) v (a ^ (a' v a))) = (a v a')
20 df-i3 46 . 2 (a ->3 a) = (((a' ^ a) v (a' ^ a')) v (a ^ (a' v a)))
2119, 20, 123tr1 63 1 (a ->3 a) = 1
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-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46
This theorem is referenced by:  bina1  282  bina2  283  ska14  514  i3orcom  525  i3ancom  526  i3th4  546
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