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Theorem mlaoml 833
Description: Mladen's OML.
Assertion
Ref Expression
mlaoml ((a == b) ^ (b == c)) =< (a == c)
Proof of Theorem mlaoml
StepHypRef Expression
1 u1lembi 720 . . . . 5 ((a ->1 b) ^ (b ->1 a)) = (a == b)
21ran 78 . . . 4 (((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) = ((a == b) ^ (b ->1 c))
3 mlalem 832 . . . 4 ((a == b) ^ (b ->1 c)) =< (a ->1 c)
42, 3bltr 138 . . 3 (((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) =< (a ->1 c)
5 ancom 74 . . . . . 6 ((b ->1 a) ^ (c ->1 b)) = ((c ->1 b) ^ (b ->1 a))
65ran 78 . . . . 5 (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c)) = (((c ->1 b) ^ (b ->1 a)) ^ (b ->1 c))
7 an32 83 . . . . 5 (((c ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) = (((c ->1 b) ^ (b ->1 c)) ^ (b ->1 a))
8 u1lembi 720 . . . . . 6 ((c ->1 b) ^ (b ->1 c)) = (c == b)
98ran 78 . . . . 5 (((c ->1 b) ^ (b ->1 c)) ^ (b ->1 a)) = ((c == b) ^ (b ->1 a))
106, 7, 93tr 65 . . . 4 (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c)) = ((c == b) ^ (b ->1 a))
11 mlalem 832 . . . 4 ((c == b) ^ (b ->1 a)) =< (c ->1 a)
1210, 11bltr 138 . . 3 (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c)) =< (c ->1 a)
134, 12le2an 169 . 2 ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c))) =< ((a ->1 c) ^ (c ->1 a))
14 an12 81 . . . . . 6 ((b ->1 a) ^ ((a ->1 b) ^ (c ->1 b))) = ((a ->1 b) ^ ((b ->1 a) ^ (c ->1 b)))
15 ancom 74 . . . . . . . 8 ((a ->1 b) ^ (b ->1 a)) = ((b ->1 a) ^ (a ->1 b))
1615ran 78 . . . . . . 7 (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b))) = (((b ->1 a) ^ (a ->1 b)) ^ ((b ->1 a) ^ (c ->1 b)))
17 id 59 . . . . . . 7 (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b))) = (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b)))
18 anandi 114 . . . . . . 7 ((b ->1 a) ^ ((a ->1 b) ^ (c ->1 b))) = (((b ->1 a) ^ (a ->1 b)) ^ ((b ->1 a) ^ (c ->1 b)))
1916, 17, 183tr1 63 . . . . . 6 (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b))) = ((b ->1 a) ^ ((a ->1 b) ^ (c ->1 b)))
20 anass 76 . . . . . 6 (((a ->1 b) ^ (b ->1 a)) ^ (c ->1 b)) = ((a ->1 b) ^ ((b ->1 a) ^ (c ->1 b)))
2114, 19, 203tr1 63 . . . . 5 (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b))) = (((a ->1 b) ^ (b ->1 a)) ^ (c ->1 b))
2221ran 78 . . . 4 ((((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b))) ^ (b ->1 c)) = ((((a ->1 b) ^ (b ->1 a)) ^ (c ->1 b)) ^ (b ->1 c))
23 anandir 115 . . . 4 ((((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 a) ^ (c ->1 b))) ^ (b ->1 c)) = ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c)))
24 an32 83 . . . 4 ((((a ->1 b) ^ (b ->1 a)) ^ (c ->1 b)) ^ (b ->1 c)) = ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (c ->1 b))
2522, 23, 243tr2 64 . . 3 ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c))) = ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (c ->1 b))
26 anass 76 . . 3 ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (c ->1 b)) = (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 c) ^ (c ->1 b)))
27 u1lembi 720 . . . 4 ((b ->1 c) ^ (c ->1 b)) = (b == c)
281, 272an 79 . . 3 (((a ->1 b) ^ (b ->1 a)) ^ ((b ->1 c) ^ (c ->1 b))) = ((a == b) ^ (b == c))
2925, 26, 283tr 65 . 2 ((((a ->1 b) ^ (b ->1 a)) ^ (b ->1 c)) ^ (((b ->1 a) ^ (c ->1 b)) ^ (b ->1 c))) = ((a == b) ^ (b == c))
30 u1lembi 720 . 2 ((a ->1 c) ^ (c ->1 a)) = (a == c)
3113, 29, 30le3tr2 141 1 ((a == b) ^ (b == c)) =< (a == c)
Colors of variables: term
Syntax hints:   =< wle 2   == tb 5   ^ wa 7   ->1 wi1 12
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  eqtr4  834  mlaconj4  844
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