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Theorem com3i 467
Description: Lemma 3(i) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
com3i.1 (a ^ (a' v b)) = (a ^ b)
Assertion
Ref Expression
com3i a C b
Proof of Theorem com3i
StepHypRef Expression
1 anor1 88 . . . . . . . 8 (a ^ b') = (a' v b)'
21con2 67 . . . . . . 7 (a ^ b')' = (a' v b)
32ran 78 . . . . . 6 ((a ^ b')' ^ a) = ((a' v b) ^ a)
4 ancom 74 . . . . . 6 ((a' v b) ^ a) = (a ^ (a' v b))
53, 4ax-r2 36 . . . . 5 ((a ^ b')' ^ a) = (a ^ (a' v b))
6 com3i.1 . . . . 5 (a ^ (a' v b)) = (a ^ b)
75, 6ax-r2 36 . . . 4 ((a ^ b')' ^ a) = (a ^ b)
87lor 70 . . 3 ((a ^ b') v ((a ^ b')' ^ a)) = ((a ^ b') v (a ^ b))
9 lea 160 . . . 4 (a ^ b') =< a
109oml2 451 . . 3 ((a ^ b') v ((a ^ b')' ^ a)) = a
11 ax-a2 31 . . 3 ((a ^ b') v (a ^ b)) = ((a ^ b) v (a ^ b'))
128, 10, 113tr2 64 . 2 a = ((a ^ b) v (a ^ b'))
1312df-c1 132 1 a C b
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-le1 130  df-le2 131  df-c1 132
This theorem is referenced by: (None)
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