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Theorem i3id 251
 Description: Identity for Kalmbach implication.
Assertion
Ref Expression
i3id (a3 a) = 1

Proof of Theorem i3id
StepHypRef Expression
1 ancom 74 . . . . . . . 8 (aa) = (aa )
2 dff 101 . . . . . . . . 9 0 = (aa )
32ax-r1 35 . . . . . . . 8 (aa ) = 0
41, 3ax-r2 36 . . . . . . 7 (aa) = 0
5 anidm 111 . . . . . . 7 (aa ) = a
64, 52or 72 . . . . . 6 ((aa) ∪ (aa )) = (0 ∪ a )
7 ax-a2 31 . . . . . 6 (0 ∪ a ) = (a ∪ 0)
86, 7ax-r2 36 . . . . 5 ((aa) ∪ (aa )) = (a ∪ 0)
9 or0 102 . . . . 5 (a ∪ 0) = a
108, 9ax-r2 36 . . . 4 ((aa) ∪ (aa )) = a
11 ax-a2 31 . . . . . . 7 (aa) = (aa )
12 df-t 41 . . . . . . . 8 1 = (aa )
1312ax-r1 35 . . . . . . 7 (aa ) = 1
1411, 13ax-r2 36 . . . . . 6 (aa) = 1
1514lan 77 . . . . 5 (a ∩ (aa)) = (a ∩ 1)
16 an1 106 . . . . 5 (a ∩ 1) = a
1715, 16ax-r2 36 . . . 4 (a ∩ (aa)) = a
1810, 172or 72 . . 3 (((aa) ∪ (aa )) ∪ (a ∩ (aa))) = (aa)
1918, 11ax-r2 36 . 2 (((aa) ∪ (aa )) ∪ (a ∩ (aa))) = (aa )
20 df-i3 46 . 2 (a3 a) = (((aa) ∪ (aa )) ∪ (a ∩ (aa)))
2119, 20, 123tr1 63 1 (a3 a) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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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