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Theorem oal1 1000
Description: Orthoarguesian law - ->1 version derived from ->1 version.
Assertion
Ref Expression
oal1 ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
Proof of Theorem oal1
StepHypRef Expression
1 oal2 999 . 2 ((c' ->2 a') ^ ((a' v b')' v ((c' ->2 a') ^ (c' ->2 b')))) =< (c' ->2 b')
2 i1i2 266 . . 3 (a ->1 c) = (c' ->2 a')
3 df-a 40 . . . 4 (a ^ b) = (a' v b')'
4 i1i2 266 . . . . 5 (b ->1 c) = (c' ->2 b')
52, 42an 79 . . . 4 ((a ->1 c) ^ (b ->1 c)) = ((c' ->2 a') ^ (c' ->2 b'))
63, 52or 72 . . 3 ((a ^ b) v ((a ->1 c) ^ (b ->1 c))) = ((a' v b')' v ((c' ->2 a') ^ (c' ->2 b')))
72, 62an 79 . 2 ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) = ((c' ->2 a') ^ ((a' v b')' v ((c' ->2 a') ^ (c' ->2 b'))))
81, 7, 4le3tr1 140 1 ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)
Colors of variables: term
Syntax hints:   =< wle 2  '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-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998
This theorem depends on definitions:  df-a 40  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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