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Theorem 3vded12 815
Description: A 3-variable theorem. Experiment with weak deduction theorem.
Hypotheses
Ref Expression
3vded12.1 (a ^ (c ->1 a)) =< (c ->1 (b ->1 a))
3vded12.2 c =< a
Assertion
Ref Expression
3vded12 c =< (b ->1 a)
Proof of Theorem 3vded12
StepHypRef Expression
1 le1 146 . . 3 (c ->1 (b ->1 a)) =< 1
2 df-t 41 . . . 4 1 = (a v a')
3 an1 106 . . . . . . . 8 (a ^ 1) = a
43ax-r1 35 . . . . . . 7 a = (a ^ 1)
5 3vded12.2 . . . . . . . . . 10 c =< a
65u1lemle1 710 . . . . . . . . 9 (c ->1 a) = 1
76lan 77 . . . . . . . 8 (a ^ (c ->1 a)) = (a ^ 1)
87ax-r1 35 . . . . . . 7 (a ^ 1) = (a ^ (c ->1 a))
94, 8ax-r2 36 . . . . . 6 a = (a ^ (c ->1 a))
10 3vded12.1 . . . . . 6 (a ^ (c ->1 a)) =< (c ->1 (b ->1 a))
119, 10bltr 138 . . . . 5 a =< (c ->1 (b ->1 a))
125lecon 154 . . . . . 6 a' =< c'
13 leo 158 . . . . . . 7 c' =< (c' v (c ^ (b ->1 a)))
14 df-i1 44 . . . . . . . 8 (c ->1 (b ->1 a)) = (c' v (c ^ (b ->1 a)))
1514ax-r1 35 . . . . . . 7 (c' v (c ^ (b ->1 a))) = (c ->1 (b ->1 a))
1613, 15lbtr 139 . . . . . 6 c' =< (c ->1 (b ->1 a))
1712, 16letr 137 . . . . 5 a' =< (c ->1 (b ->1 a))
1811, 17lel2or 170 . . . 4 (a v a') =< (c ->1 (b ->1 a))
192, 18bltr 138 . . 3 1 =< (c ->1 (b ->1 a))
201, 19lebi 145 . 2 (c ->1 (b ->1 a)) = 1
2120u1lemle2 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: (None)
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