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Theorem 3vth1 804
Description: A 3-variable theorem. Equivalent to OML.
Assertion
Ref Expression
3vth1 ((a ->2 b) ^ (b v c)') =< (a ->2 c)
Proof of Theorem 3vth1
StepHypRef Expression
1 anor3 90 . . . . . . 7 (b' ^ c') = (b v c)'
21lan 77 . . . . . 6 ((b v (b' ^ a')) ^ (b' ^ c')) = ((b v (b' ^ a')) ^ (b v c)')
32ax-r1 35 . . . . 5 ((b v (b' ^ a')) ^ (b v c)') = ((b v (b' ^ a')) ^ (b' ^ c'))
4 anass 76 . . . . . 6 (((b v (b' ^ a')) ^ b') ^ c') = ((b v (b' ^ a')) ^ (b' ^ c'))
54ax-r1 35 . . . . 5 ((b v (b' ^ a')) ^ (b' ^ c')) = (((b v (b' ^ a')) ^ b') ^ c')
63, 5ax-r2 36 . . . 4 ((b v (b' ^ a')) ^ (b v c)') = (((b v (b' ^ a')) ^ b') ^ c')
7 ancom 74 . . . . . . 7 ((b v (b' ^ a')) ^ b') = (b' ^ (b v (b' ^ a')))
8 omlan 448 . . . . . . 7 (b' ^ (b v (b' ^ a'))) = (b' ^ a')
97, 8ax-r2 36 . . . . . 6 ((b v (b' ^ a')) ^ b') = (b' ^ a')
10 lear 161 . . . . . 6 (b' ^ a') =< a'
119, 10bltr 138 . . . . 5 ((b v (b' ^ a')) ^ b') =< a'
1211leran 153 . . . 4 (((b v (b' ^ a')) ^ b') ^ c') =< (a' ^ c')
136, 12bltr 138 . . 3 ((b v (b' ^ a')) ^ (b v c)') =< (a' ^ c')
14 leor 159 . . 3 (a' ^ c') =< (c v (a' ^ c'))
1513, 14letr 137 . 2 ((b v (b' ^ a')) ^ (b v c)') =< (c v (a' ^ c'))
16 df-i2 45 . . . 4 (a ->2 b) = (b v (a' ^ b'))
17 ancom 74 . . . . 5 (a' ^ b') = (b' ^ a')
1817lor 70 . . . 4 (b v (a' ^ b')) = (b v (b' ^ a'))
1916, 18ax-r2 36 . . 3 (a ->2 b) = (b v (b' ^ a'))
2019ran 78 . 2 ((a ->2 b) ^ (b v c)') = ((b v (b' ^ a')) ^ (b v c)')
21 df-i2 45 . 2 (a ->2 c) = (c v (a' ^ c'))
2215, 20, 21le3tr1 140 1 ((a ->2 b) ^ (b v c)') =< (a ->2 c)
Colors of variables: term
Syntax hints:   =< wle 2  '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-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
This theorem is referenced by:  3vth2  805  3vth3  806
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