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Theorem 3oa3 1025
Description: 3-variable orthoarguesion law expressed with the 3OA identity abbreviation.
Assertion
Ref Expression
3oa3 ((a ->1 c) ^ (a == c ==OA b)) =< (b ->1 c)
Proof of Theorem 3oa3
StepHypRef Expression
1 df-id3oa 57 . . 3 (a == c ==OA b) = (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))
21lan 77 . 2 ((a ->1 c) ^ (a == c ==OA b)) = ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))
3 3oa2 1024 . 2 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))) =< (b ->1 c)
42, 3bltr 138 1 ((a ->1 c) ^ (a == c ==OA b)) =< (b ->1 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   == wid3oa 27
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  ax-3oa 998
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-id3oa 57  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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