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Theorem 1b 117
Description: Identity law.
Assertion
Ref Expression
1b (1 == a) = a
Proof of Theorem 1b
StepHypRef Expression
1 dfb 94 . 2 (1 == a) = ((1 ^ a) v (1' ^ a'))
2 ancom 74 . . . . 5 (1 ^ a) = (a ^ 1)
3 ancom 74 . . . . . 6 (1' ^ a') = (a' ^ 1')
4 df-f 42 . . . . . . . 8 0 = 1'
54ax-r1 35 . . . . . . 7 1' = 0
65lan 77 . . . . . 6 (a' ^ 1') = (a' ^ 0)
73, 6ax-r2 36 . . . . 5 (1' ^ a') = (a' ^ 0)
82, 72or 72 . . . 4 ((1 ^ a) v (1' ^ a')) = ((a ^ 1) v (a' ^ 0))
9 an1 106 . . . . 5 (a ^ 1) = a
10 an0 108 . . . . 5 (a' ^ 0) = 0
119, 102or 72 . . . 4 ((a ^ 1) v (a' ^ 0)) = (a v 0)
128, 11ax-r2 36 . . 3 ((1 ^ a) v (1' ^ a')) = (a v 0)
13 or0 102 . . 3 (a v 0) = a
1412, 13ax-r2 36 . 2 ((1 ^ a) v (1' ^ a')) = a
151, 14ax-r2 36 1 (1 == a) = a
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8  0wf 9
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42
This theorem is referenced by:  wr3  198  woml6  436  woml7  437  r3b  442
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