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Theorem 1oaiii 823
Description: OML analog to orthoarguesian law of Godowski/Greechie, Eq. III with ->1 instead of ->0.
Assertion
Ref Expression
1oaiii ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))

Proof of Theorem 1oaiii
StepHypRef Expression
1 anass 76 . . . . 5 (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))))
2 anidm 111 . . . . . 6 (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
32lan 77 . . . . 5 ((a ->2 b) ^ (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
41, 3ax-r2 36 . . . 4 (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
54ax-r1 35 . . 3 ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
6 1oa 820 . . . 4 ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
76leran 153 . . 3 (((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
85, 7bltr 138 . 2 ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
9 anass 76 . . . . 5 (((a ->2 c) ^ ((c v b) ->1 ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ (((c v b) ->1 ((a ->2 c) ^ (a ->2 b))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))))
10 ancom 74 . . . . . . . . . 10 ((a ->2 c) ^ (a ->2 b)) = ((a ->2 b) ^ (a ->2 c))
1110ud1lem0a 255 . . . . . . . . 9 ((c v b) ->1 ((a ->2 c) ^ (a ->2 b))) = ((c v b) ->1 ((a ->2 b) ^ (a ->2 c)))
12 ax-a2 31 . . . . . . . . . 10 (c v b) = (b v c)
1312ud1lem0b 256 . . . . . . . . 9 ((c v b) ->1 ((a ->2 b) ^ (a ->2 c))) = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
1411, 13ax-r2 36 . . . . . . . 8 ((c v b) ->1 ((a ->2 c) ^ (a ->2 b))) = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
1514ran 78 . . . . . . 7 (((c v b) ->1 ((a ->2 c) ^ (a ->2 b))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
1615, 2ax-r2 36 . . . . . 6 (((c v b) ->1 ((a ->2 c) ^ (a ->2 b))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))
1716lan 77 . . . . 5 ((a ->2 c) ^ (((c v b) ->1 ((a ->2 c) ^ (a ->2 b))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
189, 17ax-r2 36 . . . 4 (((a ->2 c) ^ ((c v b) ->1 ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
1918ax-r1 35 . . 3 ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 c) ^ ((c v b) ->1 ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
20 1oa 820 . . . 4 ((a ->2 c) ^ ((c v b) ->1 ((a ->2 c) ^ (a ->2 b)))) =< (a ->2 b)
2120leran 153 . . 3 (((a ->2 c) ^ ((c v b) ->1 ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
2219, 21bltr 138 . 2 ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
238, 22lebi 145 1 ((a ->2 b) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c) ->1 ((a ->2 b) ^ (a ->2 c))))
Colors of variables: term
Syntax hints:   = wb 1   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13
This theorem is referenced by:  1oaii  824
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
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