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Theorem lem4.6.6i3j1 1097
Description: Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 3, and j is set to 1. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem4.6.6i3j1 ((a ->3 b) v (a ->1 b)) = (a ->0 b)
Proof of Theorem lem4.6.6i3j1
StepHypRef Expression
1 ax-a3 32 . . 3 ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v (a' v (a ^ b))) = (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v (a' v (a ^ b))))
2 ax-a3 32 . . . . 5 (((a ^ (a' v b)) v a') v (a ^ b)) = ((a ^ (a' v b)) v (a' v (a ^ b)))
32ax-r1 35 . . . 4 ((a ^ (a' v b)) v (a' v (a ^ b))) = (((a ^ (a' v b)) v a') v (a ^ b))
43lor 70 . . 3 (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v (a' v (a ^ b)))) = (((a' ^ b) v (a' ^ b')) v (((a ^ (a' v b)) v a') v (a ^ b)))
5 ax-a2 31 . . . . . . 7 ((a ^ (a' v b)) v a') = (a' v (a ^ (a' v b)))
6 omln 446 . . . . . . 7 (a' v (a ^ (a' v b))) = (a' v b)
75, 6ax-r2 36 . . . . . 6 ((a ^ (a' v b)) v a') = (a' v b)
87ax-r5 38 . . . . 5 (((a ^ (a' v b)) v a') v (a ^ b)) = ((a' v b) v (a ^ b))
98lor 70 . . . 4 (((a' ^ b) v (a' ^ b')) v (((a ^ (a' v b)) v a') v (a ^ b))) = (((a' ^ b) v (a' ^ b')) v ((a' v b) v (a ^ b)))
10 leao1 162 . . . . . . 7 (a' ^ b) =< (a' v b)
11 leao1 162 . . . . . . 7 (a' ^ b') =< (a' v b)
1210, 11lel2or 170 . . . . . 6 ((a' ^ b) v (a' ^ b')) =< (a' v b)
13 leid 148 . . . . . . 7 (a' v b) =< (a' v b)
14 leao4 165 . . . . . . 7 (a ^ b) =< (a' v b)
1513, 14lel2or 170 . . . . . 6 ((a' v b) v (a ^ b)) =< (a' v b)
1612, 15lel2or 170 . . . . 5 (((a' ^ b) v (a' ^ b')) v ((a' v b) v (a ^ b))) =< (a' v b)
17 leo 158 . . . . . 6 (a' v b) =< ((a' v b) v (a ^ b))
1817lerr 150 . . . . 5 (a' v b) =< (((a' ^ b) v (a' ^ b')) v ((a' v b) v (a ^ b)))
1916, 18lebi 145 . . . 4 (((a' ^ b) v (a' ^ b')) v ((a' v b) v (a ^ b))) = (a' v b)
209, 19ax-r2 36 . . 3 (((a' ^ b) v (a' ^ b')) v (((a ^ (a' v b)) v a') v (a ^ b))) = (a' v b)
211, 4, 203tr 65 . 2 ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v (a' v (a ^ b))) = (a' v b)
22 df-i3 46 . . 3 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
23 df-i1 44 . . 3 (a ->1 b) = (a' v (a ^ b))
2422, 232or 72 . 2 ((a ->3 b) v (a ->1 b)) = ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v (a' v (a ^ b)))
25 df-i0 43 . 2 (a ->0 b) = (a' v b)
2621, 24, 253tr1 63 1 ((a ->3 b) v (a ->1 b)) = (a ->0 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->0 wi0 11   ->1 wi1 12   ->3 wi3 14
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-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i3 46  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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