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Theorem 3vth5 808
Description: A 3-variable theorem.
Assertion
Ref Expression
3vth5 ((a ->2 b)' ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))
Proof of Theorem 3vth5
StepHypRef Expression
1 ax-a3 32 . . 3 ((b v c) v ((b v (a' ^ b')) ^ (b v c)')) = (b v (c v ((b v (a' ^ b')) ^ (b v c)')))
2 or12 80 . . . 4 (b v (c v ((b v (a' ^ b')) ^ (b v c)'))) = (c v (b v ((b v (a' ^ b')) ^ (b v c)')))
3 comorr 184 . . . . . . 7 b C (b v (a' ^ b'))
4 comorr 184 . . . . . . . 8 b C (b v c)
54comcom2 183 . . . . . . 7 b C (b v c)'
63, 5fh3 471 . . . . . 6 (b v ((b v (a' ^ b')) ^ (b v c)')) = ((b v (b v (a' ^ b'))) ^ (b v (b v c)'))
7 ax-a3 32 . . . . . . . . 9 ((b v b) v (a' ^ b')) = (b v (b v (a' ^ b')))
87ax-r1 35 . . . . . . . 8 (b v (b v (a' ^ b'))) = ((b v b) v (a' ^ b'))
9 oridm 110 . . . . . . . . 9 (b v b) = b
109ax-r5 38 . . . . . . . 8 ((b v b) v (a' ^ b')) = (b v (a' ^ b'))
118, 10ax-r2 36 . . . . . . 7 (b v (b v (a' ^ b'))) = (b v (a' ^ b'))
12 ancom 74 . . . . . . . . . 10 (c' ^ b') = (b' ^ c')
13 anor3 90 . . . . . . . . . 10 (b' ^ c') = (b v c)'
1412, 13ax-r2 36 . . . . . . . . 9 (c' ^ b') = (b v c)'
1514ax-r1 35 . . . . . . . 8 (b v c)' = (c' ^ b')
1615lor 70 . . . . . . 7 (b v (b v c)') = (b v (c' ^ b'))
1711, 162an 79 . . . . . 6 ((b v (b v (a' ^ b'))) ^ (b v (b v c)')) = ((b v (a' ^ b')) ^ (b v (c' ^ b')))
186, 17ax-r2 36 . . . . 5 (b v ((b v (a' ^ b')) ^ (b v c)')) = ((b v (a' ^ b')) ^ (b v (c' ^ b')))
1918lor 70 . . . 4 (c v (b v ((b v (a' ^ b')) ^ (b v c)'))) = (c v ((b v (a' ^ b')) ^ (b v (c' ^ b'))))
202, 19ax-r2 36 . . 3 (b v (c v ((b v (a' ^ b')) ^ (b v c)'))) = (c v ((b v (a' ^ b')) ^ (b v (c' ^ b'))))
211, 20ax-r2 36 . 2 ((b v c) v ((b v (a' ^ b')) ^ (b v c)')) = (c v ((b v (a' ^ b')) ^ (b v (c' ^ b'))))
22 df-i2 45 . . 3 ((a ->2 b)' ->2 (b v c)) = ((b v c) v ((a ->2 b)'' ^ (b v c)'))
23 df-i2 45 . . . . . . . 8 (a ->2 b) = (b v (a' ^ b'))
2423ax-r1 35 . . . . . . 7 (b v (a' ^ b')) = (a ->2 b)
25 ax-a1 30 . . . . . . 7 (a ->2 b) = (a ->2 b)''
2624, 25ax-r2 36 . . . . . 6 (b v (a' ^ b')) = (a ->2 b)''
2726ran 78 . . . . 5 ((b v (a' ^ b')) ^ (b v c)') = ((a ->2 b)'' ^ (b v c)')
2827lor 70 . . . 4 ((b v c) v ((b v (a' ^ b')) ^ (b v c)')) = ((b v c) v ((a ->2 b)'' ^ (b v c)'))
2928ax-r1 35 . . 3 ((b v c) v ((a ->2 b)'' ^ (b v c)')) = ((b v c) v ((b v (a' ^ b')) ^ (b v c)'))
3022, 29ax-r2 36 . 2 ((a ->2 b)' ->2 (b v c)) = ((b v c) v ((b v (a' ^ b')) ^ (b v c)'))
31 df-i2 45 . . . 4 (c ->2 b) = (b v (c' ^ b'))
3223, 312an 79 . . 3 ((a ->2 b) ^ (c ->2 b)) = ((b v (a' ^ b')) ^ (b v (c' ^ b')))
3332lor 70 . 2 (c v ((a ->2 b) ^ (c ->2 b))) = (c v ((b v (a' ^ b')) ^ (b v (c' ^ b'))))
3421, 30, 333tr1 63 1 ((a ->2 b)' ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))
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:  3vth6  809  3vth7  810
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