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Theorem oaliii 1001
Description: Orthoarguesian law. Godowski/Greechie, Eq. III.
Assertion
Ref Expression
oaliii ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
Proof of Theorem oaliii
StepHypRef Expression
1 anass 76 . . . . 5 (((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (((b v c)' v ((a ->2 b) ^ (a ->2 c))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))
2 anidm 111 . . . . . 6 (((b v c)' v ((a ->2 b) ^ (a ->2 c))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
32lan 77 . . . . 5 ((a ->2 b) ^ (((b v c)' v ((a ->2 b) ^ (a ->2 c))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
41, 3ax-r2 36 . . . 4 (((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
54ax-r1 35 . . 3 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
6 oal2 999 . . . 4 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
76leran 153 . . 3 (((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
85, 7bltr 138 . 2 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
9 anass 76 . . . . 5 (((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ (((c v b)' v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))
10 ax-a2 31 . . . . . . . . . 10 (c v b) = (b v c)
1110ax-r4 37 . . . . . . . . 9 (c v b)' = (b v c)'
12 ancom 74 . . . . . . . . 9 ((a ->2 c) ^ (a ->2 b)) = ((a ->2 b) ^ (a ->2 c))
1311, 122or 72 . . . . . . . 8 ((c v b)' v ((a ->2 c) ^ (a ->2 b))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
1413ran 78 . . . . . . 7 (((c v b)' v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = (((b v c)' v ((a ->2 b) ^ (a ->2 c))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
1514, 2ax-r2 36 . . . . . 6 (((c v b)' v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
1615lan 77 . . . . 5 ((a ->2 c) ^ (((c v b)' v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
179, 16ax-r2 36 . . . 4 (((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
1817ax-r1 35 . . 3 ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
19 oal2 999 . . . 4 ((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))) =< (a ->2 b)
2019leran 153 . . 3 (((a ->2 c) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
2118, 20bltr 138 . 2 ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
228, 21lebi 145 1 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  oalii  1002  oath1  1004
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