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Theorem 3vth7 810
Description: A 3-variable theorem.
Assertion
Ref Expression
3vth7 ((a ->2 b)' ->2 (b v c)) = (a ->2 (b v c))
Proof of Theorem 3vth7
StepHypRef Expression
1 df-i2 45 . . . . 5 (a ->2 b) = (b v (a' ^ b'))
2 df-i2 45 . . . . 5 (c ->2 b) = (b v (c' ^ b'))
31, 22an 79 . . . 4 ((a ->2 b) ^ (c ->2 b)) = ((b v (a' ^ b')) ^ (b v (c' ^ b')))
4 anass 76 . . . . . . . . . 10 ((a' ^ b') ^ c') = (a' ^ (b' ^ c'))
54ax-r1 35 . . . . . . . . 9 (a' ^ (b' ^ c')) = ((a' ^ b') ^ c')
6 anor3 90 . . . . . . . . . 10 (b' ^ c') = (b v c)'
76lan 77 . . . . . . . . 9 (a' ^ (b' ^ c')) = (a' ^ (b v c)')
8 an32 83 . . . . . . . . 9 ((a' ^ b') ^ c') = ((a' ^ c') ^ b')
95, 7, 83tr2 64 . . . . . . . 8 (a' ^ (b v c)') = ((a' ^ c') ^ b')
10 anidm 111 . . . . . . . . . 10 (b' ^ b') = b'
1110lan 77 . . . . . . . . 9 ((a' ^ c') ^ (b' ^ b')) = ((a' ^ c') ^ b')
1211ax-r1 35 . . . . . . . 8 ((a' ^ c') ^ b') = ((a' ^ c') ^ (b' ^ b'))
13 an4 86 . . . . . . . 8 ((a' ^ c') ^ (b' ^ b')) = ((a' ^ b') ^ (c' ^ b'))
149, 12, 133tr 65 . . . . . . 7 (a' ^ (b v c)') = ((a' ^ b') ^ (c' ^ b'))
1514lor 70 . . . . . 6 (b v (a' ^ (b v c)')) = (b v ((a' ^ b') ^ (c' ^ b')))
16 comanr2 465 . . . . . . . 8 b' C (a' ^ b')
1716comcom6 459 . . . . . . 7 b C (a' ^ b')
18 comanr2 465 . . . . . . . 8 b' C (c' ^ b')
1918comcom6 459 . . . . . . 7 b C (c' ^ b')
2017, 19fh3 471 . . . . . 6 (b v ((a' ^ b') ^ (c' ^ b'))) = ((b v (a' ^ b')) ^ (b v (c' ^ b')))
2115, 20ax-r2 36 . . . . 5 (b v (a' ^ (b v c)')) = ((b v (a' ^ b')) ^ (b v (c' ^ b')))
2221ax-r1 35 . . . 4 ((b v (a' ^ b')) ^ (b v (c' ^ b'))) = (b v (a' ^ (b v c)'))
233, 22ax-r2 36 . . 3 ((a ->2 b) ^ (c ->2 b)) = (b v (a' ^ (b v c)'))
2423lor 70 . 2 (c v ((a ->2 b) ^ (c ->2 b))) = (c v (b v (a' ^ (b v c)')))
25 3vth5 808 . 2 ((a ->2 b)' ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))
26 df-i2 45 . . 3 (a ->2 (b v c)) = ((b v c) v (a' ^ (b v c)'))
27 ax-a3 32 . . 3 ((b v c) v (a' ^ (b v c)')) = (b v (c v (a' ^ (b v c)')))
28 or12 80 . . 3 (b v (c v (a' ^ (b v c)'))) = (c v (b v (a' ^ (b v c)')))
2926, 27, 283tr 65 . 2 (a ->2 (b v c)) = (c v (b v (a' ^ (b v c)')))
3024, 25, 293tr1 63 1 ((a ->2 b)' ->2 (b v c)) = (a ->2 (b v c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  3vth8  811
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