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Theorem ka4lemo 228
Description: Lemma for KA4 soundness (OR version) - uses OL only.
Assertion
Ref Expression
ka4lemo ((a v b) v ((a v c) == (b v c))) = 1
Proof of Theorem ka4lemo
StepHypRef Expression
1 le1 146 . 2 ((a v b) v ((a v c) == (b v c))) =< 1
2 df-t 41 . . 3 1 = (((a v b) v c) v ((a v b) v c)')
3 leo 158 . . . . . . 7 c =< (c v (a ^ b))
4 ax-a2 31 . . . . . . 7 (c v (a ^ b)) = ((a ^ b) v c)
53, 4lbtr 139 . . . . . 6 c =< ((a ^ b) v c)
65lelor 166 . . . . 5 ((a v b) v c) =< ((a v b) v ((a ^ b) v c))
76leror 152 . . . 4 (((a v b) v c) v ((a v b) v c)') =< (((a v b) v ((a ^ b) v c)) v ((a v b) v c)')
8 ax-a3 32 . . . . 5 (((a v b) v ((a ^ b) v c)) v ((a v b) v c)') = ((a v b) v (((a ^ b) v c) v ((a v b) v c)'))
9 ledio 176 . . . . . . . . 9 (c v (a ^ b)) =< ((c v a) ^ (c v b))
10 ax-a2 31 . . . . . . . . 9 ((a ^ b) v c) = (c v (a ^ b))
11 ax-a2 31 . . . . . . . . . 10 (a v c) = (c v a)
12 ax-a2 31 . . . . . . . . . 10 (b v c) = (c v b)
1311, 122an 79 . . . . . . . . 9 ((a v c) ^ (b v c)) = ((c v a) ^ (c v b))
149, 10, 13le3tr1 140 . . . . . . . 8 ((a ^ b) v c) =< ((a v c) ^ (b v c))
1514leror 152 . . . . . . 7 (((a ^ b) v c) v ((a v b) v c)') =< (((a v c) ^ (b v c)) v ((a v b) v c)')
16 dfb 94 . . . . . . . . 9 ((a v c) == (b v c)) = (((a v c) ^ (b v c)) v ((a v c)' ^ (b v c)'))
17 oran 87 . . . . . . . . . . . . 13 (a v c) = (a' ^ c')'
1817con2 67 . . . . . . . . . . . 12 (a v c)' = (a' ^ c')
19 oran 87 . . . . . . . . . . . . 13 (b v c) = (b' ^ c')'
2019con2 67 . . . . . . . . . . . 12 (b v c)' = (b' ^ c')
2118, 202an 79 . . . . . . . . . . 11 ((a v c)' ^ (b v c)') = ((a' ^ c') ^ (b' ^ c'))
22 anor1 88 . . . . . . . . . . . 12 ((a' ^ b') ^ c') = ((a' ^ b')' v c)'
23 anandir 115 . . . . . . . . . . . . 13 ((a' ^ b') ^ c') = ((a' ^ c') ^ (b' ^ c'))
2423ax-r1 35 . . . . . . . . . . . 12 ((a' ^ c') ^ (b' ^ c')) = ((a' ^ b') ^ c')
25 oran 87 . . . . . . . . . . . . . 14 (a v b) = (a' ^ b')'
2625ax-r5 38 . . . . . . . . . . . . 13 ((a v b) v c) = ((a' ^ b')' v c)
2726ax-r4 37 . . . . . . . . . . . 12 ((a v b) v c)' = ((a' ^ b')' v c)'
2822, 24, 273tr1 63 . . . . . . . . . . 11 ((a' ^ c') ^ (b' ^ c')) = ((a v b) v c)'
2921, 28ax-r2 36 . . . . . . . . . 10 ((a v c)' ^ (b v c)') = ((a v b) v c)'
3029lor 70 . . . . . . . . 9 (((a v c) ^ (b v c)) v ((a v c)' ^ (b v c)')) = (((a v c) ^ (b v c)) v ((a v b) v c)')
3116, 30ax-r2 36 . . . . . . . 8 ((a v c) == (b v c)) = (((a v c) ^ (b v c)) v ((a v b) v c)')
3231ax-r1 35 . . . . . . 7 (((a v c) ^ (b v c)) v ((a v b) v c)') = ((a v c) == (b v c))
3315, 32lbtr 139 . . . . . 6 (((a ^ b) v c) v ((a v b) v c)') =< ((a v c) == (b v c))
3433lelor 166 . . . . 5 ((a v b) v (((a ^ b) v c) v ((a v b) v c)')) =< ((a v b) v ((a v c) == (b v c)))
358, 34bltr 138 . . . 4 (((a v b) v ((a ^ b) v c)) v ((a v b) v c)') =< ((a v b) v ((a v c) == (b v c)))
367, 35letr 137 . . 3 (((a v b) v c) v ((a v b) v c)') =< ((a v b) v ((a v c) == (b v c)))
372, 36bltr 138 . 2 1 =< ((a v b) v ((a v c) == (b v c)))
381, 37lebi 145 1 ((a v b) v ((a v c) == (b v c))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by:  ka4lem  229  ka4ot  435
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