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Theorem 3vded11 814
Description: A 3-variable theorem. Experiment with weak deduction theorem.
Hypothesis
Ref Expression
3vded11.1 b =< (c ->1 (b ->1 a))
Assertion
Ref Expression
3vded11 c =< (b ->1 a)
Proof of Theorem 3vded11
StepHypRef Expression
1 le1 146 . . 3 (c ->1 (b ->1 a)) =< 1
2 df-t 41 . . . . 5 1 = ((b v c') v (b v c')')
3 ancom 74 . . . . . . . 8 (c ^ b') = (b' ^ c)
4 anor2 89 . . . . . . . 8 (b' ^ c) = (b v c')'
53, 4ax-r2 36 . . . . . . 7 (c ^ b') = (b v c')'
65lor 70 . . . . . 6 ((b v c') v (c ^ b')) = ((b v c') v (b v c')')
76ax-r1 35 . . . . 5 ((b v c') v (b v c')') = ((b v c') v (c ^ b'))
8 ax-a3 32 . . . . 5 ((b v c') v (c ^ b')) = (b v (c' v (c ^ b')))
92, 7, 83tr 65 . . . 4 1 = (b v (c' v (c ^ b')))
10 3vded11.1 . . . . 5 b =< (c ->1 (b ->1 a))
11 leo 158 . . . . . . . . 9 b' =< (b' v (b ^ a))
12 df-i1 44 . . . . . . . . . 10 (b ->1 a) = (b' v (b ^ a))
1312ax-r1 35 . . . . . . . . 9 (b' v (b ^ a)) = (b ->1 a)
1411, 13lbtr 139 . . . . . . . 8 b' =< (b ->1 a)
1514lelan 167 . . . . . . 7 (c ^ b') =< (c ^ (b ->1 a))
1615lelor 166 . . . . . 6 (c' v (c ^ b')) =< (c' v (c ^ (b ->1 a)))
17 df-i1 44 . . . . . . 7 (c ->1 (b ->1 a)) = (c' v (c ^ (b ->1 a)))
1817ax-r1 35 . . . . . 6 (c' v (c ^ (b ->1 a))) = (c ->1 (b ->1 a))
1916, 18lbtr 139 . . . . 5 (c' v (c ^ b')) =< (c ->1 (b ->1 a))
2010, 19lel2or 170 . . . 4 (b v (c' v (c ^ b'))) =< (c ->1 (b ->1 a))
219, 20bltr 138 . . 3 1 =< (c ->1 (b ->1 a))
221, 21lebi 145 . 2 (c ->1 (b ->1 a)) = 1
2322u1lemle2 715 1 c =< (b ->1 a)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8   ->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:  3vded13  816
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