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Theorem wcom3ii 419
Description: Lemma 3(ii) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomcom.1 C (a, b) = 1
Assertion
Ref Expression
wcom3ii ((a ^ (a' v b)) == (a ^ b)) = 1
Proof of Theorem wcom3ii
StepHypRef Expression
1 wcomcom.1 . . . . . 6 C (a, b) = 1
21wcomcom 414 . . . . 5 C (b, a) = 1
32wcomd 418 . . . 4 (b == ((b v a) ^ (b v a'))) = 1
43wlan 370 . . 3 ((a ^ b) == (a ^ ((b v a) ^ (b v a')))) = 1
5 anass 76 . . . . . 6 ((a ^ (b v a)) ^ (b v a')) = (a ^ ((b v a) ^ (b v a')))
65bi1 118 . . . . 5 (((a ^ (b v a)) ^ (b v a')) == (a ^ ((b v a) ^ (b v a')))) = 1
76wr1 197 . . . 4 ((a ^ ((b v a) ^ (b v a'))) == ((a ^ (b v a)) ^ (b v a'))) = 1
8 ax-a2 31 . . . . . . . 8 (b v a) = (a v b)
98bi1 118 . . . . . . 7 ((b v a) == (a v b)) = 1
109wlan 370 . . . . . 6 ((a ^ (b v a)) == (a ^ (a v b))) = 1
11 anabs 121 . . . . . . 7 (a ^ (a v b)) = a
1211bi1 118 . . . . . 6 ((a ^ (a v b)) == a) = 1
1310, 12wr2 371 . . . . 5 ((a ^ (b v a)) == a) = 1
14 ax-a2 31 . . . . . 6 (b v a') = (a' v b)
1514bi1 118 . . . . 5 ((b v a') == (a' v b)) = 1
1613, 15w2an 373 . . . 4 (((a ^ (b v a)) ^ (b v a')) == (a ^ (a' v b))) = 1
177, 16wr2 371 . . 3 ((a ^ ((b v a) ^ (b v a'))) == (a ^ (a' v b))) = 1
184, 17wr2 371 . 2 ((a ^ b) == (a ^ (a' v b))) = 1
1918wr1 197 1 ((a ^ (a' v b)) == (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:  wfh1  423  wfh2  424
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