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Theorem 2oath1i1 827
Description: Orthoarguesian-like OM law.
Assertion
Ref Expression
2oath1i1 ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
Proof of Theorem 2oath1i1
StepHypRef Expression
1 2oath1 826 . 2 ((c' ->2 a') ^ ((a' v b') ->2 ((c' ->2 a') ^ (c' ->2 b')))) = ((c' ->2 a') ^ (c' ->2 b'))
2 i1i2 266 . . 3 (a ->1 c) = (c' ->2 a')
3 i1i2 266 . . . . . 6 (b ->1 c) = (c' ->2 b')
42, 32an 79 . . . . 5 ((a ->1 c) ^ (b ->1 c)) = ((c' ->2 a') ^ (c' ->2 b'))
54ud2lem0a 258 . . . 4 ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))) = ((a ^ b)' ->2 ((c' ->2 a') ^ (c' ->2 b')))
6 oran3 93 . . . . . 6 (a' v b') = (a ^ b)'
76ax-r1 35 . . . . 5 (a ^ b)' = (a' v b')
87ud2lem0b 259 . . . 4 ((a ^ b)' ->2 ((c' ->2 a') ^ (c' ->2 b'))) = ((a' v b') ->2 ((c' ->2 a') ^ (c' ->2 b')))
95, 8ax-r2 36 . . 3 ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c))) = ((a' v b') ->2 ((c' ->2 a') ^ (c' ->2 b')))
102, 92an 79 . 2 ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((c' ->2 a') ^ ((a' v b') ->2 ((c' ->2 a') ^ (c' ->2 b'))))
111, 10, 43tr1 63 1 ((a ->1 c) ^ ((a ^ b)' ->2 ((a ->1 c) ^ (b ->1 c)))) = ((a ->1 c) ^ (b ->1 c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  1oath1i1u  828  d3oa  995
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