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Theorem wcom2or 427
Description: Th. 4.2 Beran p. 49.
Hypotheses
Ref Expression
wfh.1 C (a, b) = 1
wfh.2 C (a, c) = 1
Assertion
Ref Expression
wcom2or C (a, (b v c)) = 1
Proof of Theorem wcom2or
StepHypRef Expression
1 wfh.1 . . . . . . . . 9 C (a, b) = 1
21wcomcom 414 . . . . . . . 8 C (b, a) = 1
32wdf-c2 384 . . . . . . 7 (b == ((b ^ a) v (b ^ a'))) = 1
4 ancom 74 . . . . . . . . 9 (b ^ a) = (a ^ b)
5 ancom 74 . . . . . . . . 9 (b ^ a') = (a' ^ b)
64, 52or 72 . . . . . . . 8 ((b ^ a) v (b ^ a')) = ((a ^ b) v (a' ^ b))
76bi1 118 . . . . . . 7 (((b ^ a) v (b ^ a')) == ((a ^ b) v (a' ^ b))) = 1
83, 7wr2 371 . . . . . 6 (b == ((a ^ b) v (a' ^ b))) = 1
9 wfh.2 . . . . . . . . 9 C (a, c) = 1
109wcomcom 414 . . . . . . . 8 C (c, a) = 1
1110wdf-c2 384 . . . . . . 7 (c == ((c ^ a) v (c ^ a'))) = 1
12 ancom 74 . . . . . . . . 9 (c ^ a) = (a ^ c)
13 ancom 74 . . . . . . . . 9 (c ^ a') = (a' ^ c)
1412, 132or 72 . . . . . . . 8 ((c ^ a) v (c ^ a')) = ((a ^ c) v (a' ^ c))
1514bi1 118 . . . . . . 7 (((c ^ a) v (c ^ a')) == ((a ^ c) v (a' ^ c))) = 1
1611, 15wr2 371 . . . . . 6 (c == ((a ^ c) v (a' ^ c))) = 1
178, 16w2or 372 . . . . 5 ((b v c) == (((a ^ b) v (a' ^ b)) v ((a ^ c) v (a' ^ c)))) = 1
18 or4 84 . . . . . 6 (((a ^ b) v (a' ^ b)) v ((a ^ c) v (a' ^ c))) = (((a ^ b) v (a ^ c)) v ((a' ^ b) v (a' ^ c)))
1918bi1 118 . . . . 5 ((((a ^ b) v (a' ^ b)) v ((a ^ c) v (a' ^ c))) == (((a ^ b) v (a ^ c)) v ((a' ^ b) v (a' ^ c)))) = 1
2017, 19wr2 371 . . . 4 ((b v c) == (((a ^ b) v (a ^ c)) v ((a' ^ b) v (a' ^ c)))) = 1
21 ancom 74 . . . . . . . 8 ((b v c) ^ a) = (a ^ (b v c))
2221bi1 118 . . . . . . 7 (((b v c) ^ a) == (a ^ (b v c))) = 1
231, 9wfh1 423 . . . . . . 7 ((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1
2422, 23wr2 371 . . . . . 6 (((b v c) ^ a) == ((a ^ b) v (a ^ c))) = 1
25 ancom 74 . . . . . . . 8 ((b v c) ^ a') = (a' ^ (b v c))
2625bi1 118 . . . . . . 7 (((b v c) ^ a') == (a' ^ (b v c))) = 1
271wcomcom3 416 . . . . . . . 8 C (a', b) = 1
289wcomcom3 416 . . . . . . . 8 C (a', c) = 1
2927, 28wfh1 423 . . . . . . 7 ((a' ^ (b v c)) == ((a' ^ b) v (a' ^ c))) = 1
3026, 29wr2 371 . . . . . 6 (((b v c) ^ a') == ((a' ^ b) v (a' ^ c))) = 1
3124, 30w2or 372 . . . . 5 ((((b v c) ^ a) v ((b v c) ^ a')) == (((a ^ b) v (a ^ c)) v ((a' ^ b) v (a' ^ c)))) = 1
3231wr1 197 . . . 4 ((((a ^ b) v (a ^ c)) v ((a' ^ b) v (a' ^ c))) == (((b v c) ^ a) v ((b v c) ^ a'))) = 1
3320, 32wr2 371 . . 3 ((b v c) == (((b v c) ^ a) v ((b v c) ^ a'))) = 1
3433wdf-c1 383 . 2 C ((b v c), a) = 1
3534wcomcom 414 1 C (a, (b v c)) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   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:  wcom2an  428  ska2  432  ska4  433
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