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Theorem elimcons 868
Description: Consequent elimination law.
Hypotheses
Ref Expression
elimcons.1 (a ->1 c) = (b ->1 c)
elimcons.2 (a ^ c) =< (b v c')
Assertion
Ref Expression
elimcons a =< b
Proof of Theorem elimcons
StepHypRef Expression
1 df-t 41 . . . . . . . 8 1 = (a v a')
2 elimcons.1 . . . . . . . . . 10 (a ->1 c) = (b ->1 c)
3 elimcons.2 . . . . . . . . . 10 (a ^ c) =< (b v c')
42, 3elimconslem 867 . . . . . . . . 9 a =< (b v c')
54leror 152 . . . . . . . 8 (a v a') =< ((b v c') v a')
61, 5bltr 138 . . . . . . 7 1 =< ((b v c') v a')
76lelan 167 . . . . . 6 (b' ^ 1) =< (b' ^ ((b v c') v a'))
8 an1 106 . . . . . 6 (b' ^ 1) = b'
9 comor1 461 . . . . . . . 8 (b v c') C b
109comcom2 183 . . . . . . 7 (b v c') C b'
114lecom 180 . . . . . . . . 9 a C (b v c')
1211comcom3 454 . . . . . . . 8 a' C (b v c')
1312comcom 453 . . . . . . 7 (b v c') C a'
1410, 13fh2 470 . . . . . 6 (b' ^ ((b v c') v a')) = ((b' ^ (b v c')) v (b' ^ a'))
157, 8, 14le3tr2 141 . . . . 5 b' =< ((b' ^ (b v c')) v (b' ^ a'))
162negant 852 . . . . . . . . . . 11 (a' ->1 c) = (b' ->1 c)
17 df-i1 44 . . . . . . . . . . 11 (a' ->1 c) = (a'' v (a' ^ c))
18 df-i1 44 . . . . . . . . . . 11 (b' ->1 c) = (b'' v (b' ^ c))
1916, 17, 183tr2 64 . . . . . . . . . 10 (a'' v (a' ^ c)) = (b'' v (b' ^ c))
20 anor2 89 . . . . . . . . . . 11 (a' ^ c) = (a v c')'
2120lor 70 . . . . . . . . . 10 (a'' v (a' ^ c)) = (a'' v (a v c')')
22 anor2 89 . . . . . . . . . . 11 (b' ^ c) = (b v c')'
2322lor 70 . . . . . . . . . 10 (b'' v (b' ^ c)) = (b'' v (b v c')')
2419, 21, 233tr2 64 . . . . . . . . 9 (a'' v (a v c')') = (b'' v (b v c')')
2524ax-r1 35 . . . . . . . 8 (b'' v (b v c')') = (a'' v (a v c')')
2625ax-r4 37 . . . . . . 7 (b'' v (b v c')')' = (a'' v (a v c')')'
27 df-a 40 . . . . . . 7 (b' ^ (b v c')) = (b'' v (b v c')')'
28 df-a 40 . . . . . . 7 (a' ^ (a v c')) = (a'' v (a v c')')'
2926, 27, 283tr1 63 . . . . . 6 (b' ^ (b v c')) = (a' ^ (a v c'))
3029ax-r5 38 . . . . 5 ((b' ^ (b v c')) v (b' ^ a')) = ((a' ^ (a v c')) v (b' ^ a'))
3115, 30lbtr 139 . . . 4 b' =< ((a' ^ (a v c')) v (b' ^ a'))
32 lear 161 . . . . 5 (b' ^ a') =< a'
3332lelor 166 . . . 4 ((a' ^ (a v c')) v (b' ^ a')) =< ((a' ^ (a v c')) v a')
3431, 33letr 137 . . 3 b' =< ((a' ^ (a v c')) v a')
35 lea 160 . . . 4 (a' ^ (a v c')) =< a'
3635df-le2 131 . . 3 ((a' ^ (a v c')) v a') = a'
3734, 36lbtr 139 . 2 b' =< a'
3837lecon1 155 1 a =< b
Colors of variables: term
Syntax hints:   = wb 1   =< 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:  elimcons2  869
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