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Theorem 3oa2 1024
Description: Alternate form for the 3-variable orthoarguesion law.
Assertion
Ref Expression
3oa2 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))) =< (b ->1 c)

Proof of Theorem 3oa2
StepHypRef Expression
1 ax-3oa 998 . 2 (((a' ->1 c) ->1 c) ^ (((a' ->1 c) ^ (b' ->1 c)) v (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)))) =< ((b' ->1 c) ->1 c)
2 u1lem11 780 . . 3 ((a' ->1 c) ->1 c) = (a ->1 c)
3 ax-a2 31 . . . 4 (((a' ->1 c) ^ (b' ->1 c)) v (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c))) = ((((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))
4 u1lem11 780 . . . . . 6 ((b' ->1 c) ->1 c) = (b ->1 c)
52, 42an 79 . . . . 5 (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)) = ((a ->1 c) ^ (b ->1 c))
65ax-r5 38 . . . 4 ((((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))) = (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))
73, 6ax-r2 36 . . 3 (((a' ->1 c) ^ (b' ->1 c)) v (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c))) = (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c)))
82, 72an 79 . 2 (((a' ->1 c) ->1 c) ^ (((a' ->1 c) ^ (b' ->1 c)) v (((a' ->1 c) ->1 c) ^ ((b' ->1 c) ->1 c)))) = ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) v ((a' ->1 c) ^ (b' ->1 c))))
91, 8, 4le3tr2 141 1 ((a ->1 c) ^ (((a ->1 c) ^ (b ->1 c)) 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
This theorem is referenced by:  3oa3  1025
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-le1 130  df-le2 131  df-c1 132  df-c2 133
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