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Theorem wcom3i 422
Description: Lemma 3(i) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcom3i.1 ((a ^ (a' v b)) == (a ^ b)) = 1
Assertion
Ref Expression
wcom3i C (a, b) = 1
Proof of Theorem wcom3i
StepHypRef Expression
1 anor1 88 . . . . . . . . 9 (a ^ b') = (a' v b)'
21bi1 118 . . . . . . . 8 ((a ^ b') == (a' v b)') = 1
32wcon2 208 . . . . . . 7 ((a ^ b')' == (a' v b)) = 1
43wran 369 . . . . . 6 (((a ^ b')' ^ a) == ((a' v b) ^ a)) = 1
5 ancom 74 . . . . . . 7 ((a' v b) ^ a) = (a ^ (a' v b))
65bi1 118 . . . . . 6 (((a' v b) ^ a) == (a ^ (a' v b))) = 1
74, 6wr2 371 . . . . 5 (((a ^ b')' ^ a) == (a ^ (a' v b))) = 1
8 wcom3i.1 . . . . 5 ((a ^ (a' v b)) == (a ^ b)) = 1
97, 8wr2 371 . . . 4 (((a ^ b')' ^ a) == (a ^ b)) = 1
109wlor 368 . . 3 (((a ^ b') v ((a ^ b')' ^ a)) == ((a ^ b') v (a ^ b))) = 1
11 wlea 388 . . . 4 ((a ^ b') =<2 a) = 1
1211wom4 380 . . 3 (((a ^ b') v ((a ^ b')' ^ a)) == a) = 1
13 ax-a2 31 . . . 4 ((a ^ b') v (a ^ b)) = ((a ^ b) v (a ^ b'))
1413bi1 118 . . 3 (((a ^ b') v (a ^ b)) == ((a ^ b) v (a ^ b'))) = 1
1510, 12, 14w3tr2 375 . 2 (a == ((a ^ b) v (a ^ b'))) = 1
1615wdf-c1 383 1 C (a, b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8   C wcmtr 29
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134
This theorem is referenced by: (None)
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