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Theorem mloa 1018
Description: Mladen's OA
Assertion
Ref Expression
mloa ((a == b) ^ ((b == c) v ((b v (a == b)) ^ (c v (a == c))))) =< (c v (a == c))
Proof of Theorem mloa
StepHypRef Expression
1 lea 160 . . . 4 ((a ->2 b) ^ (b ->2 a)) =< (a ->2 b)
2 ax-a3 32 . . . . . 6 (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c))) = ((b ^ c) v ((b' ^ c') v ((a ->2 b) ^ (a ->2 c))))
3 or12 80 . . . . . . 7 ((b ^ c) v ((b' ^ c') v ((a ->2 b) ^ (a ->2 c)))) = ((b' ^ c') v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
4 anor3 90 . . . . . . . 8 (b' ^ c') = (b v c)'
54ax-r5 38 . . . . . . 7 ((b' ^ c') v ((b ^ c) v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)' v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
63, 5ax-r2 36 . . . . . 6 ((b ^ c) v ((b' ^ c') v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)' v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
72, 6ax-r2 36 . . . . 5 (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
8 leo 158 . . . . . . . . 9 b =< (b v (a' ^ b'))
9 df-i2 45 . . . . . . . . . 10 (a ->2 b) = (b v (a' ^ b'))
109ax-r1 35 . . . . . . . . 9 (b v (a' ^ b')) = (a ->2 b)
118, 10lbtr 139 . . . . . . . 8 b =< (a ->2 b)
12 leo 158 . . . . . . . . 9 c =< (c v (a' ^ c'))
13 df-i2 45 . . . . . . . . . 10 (a ->2 c) = (c v (a' ^ c'))
1413ax-r1 35 . . . . . . . . 9 (c v (a' ^ c')) = (a ->2 c)
1512, 14lbtr 139 . . . . . . . 8 c =< (a ->2 c)
1611, 15le2an 169 . . . . . . 7 (b ^ c) =< ((a ->2 b) ^ (a ->2 c))
17 id 59 . . . . . . . 8 ((a ->2 b) ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
1817bile 142 . . . . . . 7 ((a ->2 b) ^ (a ->2 c)) =< ((a ->2 b) ^ (a ->2 c))
1916, 18lel2or 170 . . . . . 6 ((b ^ c) v ((a ->2 b) ^ (a ->2 c))) =< ((a ->2 b) ^ (a ->2 c))
2019lelor 166 . . . . 5 ((b v c)' v ((b ^ c) v ((a ->2 b) ^ (a ->2 c)))) =< ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
217, 20bltr 138 . . . 4 (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c))) =< ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
221, 21le2an 169 . . 3 (((a ->2 b) ^ (b ->2 a)) ^ (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
23 oal2 999 . . 3 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
2422, 23letr 137 . 2 (((a ->2 b) ^ (b ->2 a)) ^ (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
25 u2lembi 721 . . 3 ((a ->2 b) ^ (b ->2 a)) = (a == b)
26 dfb 94 . . . . 5 (b == c) = ((b ^ c) v (b' ^ c'))
2726ax-r1 35 . . . 4 ((b ^ c) v (b' ^ c')) = (b == c)
28 i2bi 722 . . . . 5 (a ->2 b) = (b v (a == b))
29 i2bi 722 . . . . 5 (a ->2 c) = (c v (a == c))
3028, 292an 79 . . . 4 ((a ->2 b) ^ (a ->2 c)) = ((b v (a == b)) ^ (c v (a == c)))
3127, 302or 72 . . 3 (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c))) = ((b == c) v ((b v (a == b)) ^ (c v (a == c))))
3225, 312an 79 . 2 (((a ->2 b) ^ (b ->2 a)) ^ (((b ^ c) v (b' ^ c')) v ((a ->2 b) ^ (a ->2 c)))) = ((a == b) ^ ((b == c) v ((b v (a == b)) ^ (c v (a == c)))))
3324, 32, 29le3tr2 141 1 ((a == b) ^ ((b == c) v ((b v (a == b)) ^ (c v (a == c))))) =< (c v (a == c))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   == tb 5   v wo 6   ^ wa 7   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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