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Theorem i3th5 547
Description: Theorem for Kalmbach implication.
Assertion
Ref Expression
i3th5 ((a ->3 b) ->3 (a ->3 (a ->3 b))) = 1
Proof of Theorem i3th5
StepHypRef Expression
1 ax-a2 31 . . . . . 6 ((a' ^ b) v (a' ^ b')) = ((a' ^ b') v (a' ^ b))
2 lea 160 . . . . . . 7 (a' ^ b') =< a'
3 lear 161 . . . . . . 7 (a' ^ b) =< b
42, 3le2or 168 . . . . . 6 ((a' ^ b') v (a' ^ b)) =< (a' v b)
51, 4bltr 138 . . . . 5 ((a' ^ b) v (a' ^ b')) =< (a' v b)
6 lear 161 . . . . 5 (a ^ (a' v b)) =< (a' v b)
75, 6le2or 168 . . . 4 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) =< ((a' v b) v (a' v b))
8 oridm 110 . . . 4 ((a' v b) v (a' v b)) = (a' v b)
97, 8lbtr 139 . . 3 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) =< (a' v b)
10 df-i3 46 . . 3 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
11 lem4 511 . . 3 (a ->3 (a ->3 b)) = (a' v b)
129, 10, 11le3tr1 140 . 2 (a ->3 b) =< (a ->3 (a ->3 b))
1312lei3 246 1 ((a ->3 b) ->3 (a ->3 (a ->3 b))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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