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Theorem comm1 179
Description: Commutation with 1. Kalmbach 83 p. 20.
Assertion
Ref Expression
comm1 1 C a
Proof of Theorem comm1
StepHypRef Expression
1 df-t 41 . . 3 1 = (a v a')
2 ancom 74 . . . . . 6 (1 ^ a) = (a ^ 1)
3 an1 106 . . . . . 6 (a ^ 1) = a
42, 3ax-r2 36 . . . . 5 (1 ^ a) = a
5 ancom 74 . . . . . 6 (1 ^ a') = (a' ^ 1)
6 an1 106 . . . . . 6 (a' ^ 1) = a'
75, 6ax-r2 36 . . . . 5 (1 ^ a') = a'
84, 72or 72 . . . 4 ((1 ^ a) v (1 ^ a')) = (a v a')
98ax-r1 35 . . 3 (a v a') = ((1 ^ a) v (1 ^ a'))
101, 9ax-r2 36 . 2 1 = ((1 ^ a) v (1 ^ a'))
1110df-c1 132 1 1 C a
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7  1wt 8
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-c1 132
This theorem is referenced by:  wcom1  408
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