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Theorem negantlem9 859
Description: Negated antecedent identity.
Hypothesis
Ref Expression
negant.1 (a ->1 c) = (b ->1 c)
Assertion
Ref Expression
negantlem9 (a ->3 c) =< (b ->3 c)
Proof of Theorem negantlem9
StepHypRef Expression
1 leao4 165 . . . . 5 (a' ^ c) =< (b' v c)
2 leor 159 . . . . . 6 (a' ^ c) =< (a v (a' ^ c))
3 negant.1 . . . . . . . . 9 (a ->1 c) = (b ->1 c)
43sac 835 . . . . . . . 8 (a' ->1 c) = (b' ->1 c)
5 df-i1 44 . . . . . . . . 9 (a' ->1 c) = (a'' v (a' ^ c))
6 ax-a1 30 . . . . . . . . . . 11 a = a''
76ax-r5 38 . . . . . . . . . 10 (a v (a' ^ c)) = (a'' v (a' ^ c))
87ax-r1 35 . . . . . . . . 9 (a'' v (a' ^ c)) = (a v (a' ^ c))
95, 8ax-r2 36 . . . . . . . 8 (a' ->1 c) = (a v (a' ^ c))
10 df-i1 44 . . . . . . . . 9 (b' ->1 c) = (b'' v (b' ^ c))
11 ax-a1 30 . . . . . . . . . . 11 b = b''
1211ax-r5 38 . . . . . . . . . 10 (b v (b' ^ c)) = (b'' v (b' ^ c))
1312ax-r1 35 . . . . . . . . 9 (b'' v (b' ^ c)) = (b v (b' ^ c))
1410, 13ax-r2 36 . . . . . . . 8 (b' ->1 c) = (b v (b' ^ c))
154, 9, 143tr2 64 . . . . . . 7 (a v (a' ^ c)) = (b v (b' ^ c))
16 leo 158 . . . . . . . 8 b =< (b v (b' ^ c'))
1716leror 152 . . . . . . 7 (b v (b' ^ c)) =< ((b v (b' ^ c')) v (b' ^ c))
1815, 17bltr 138 . . . . . 6 (a v (a' ^ c)) =< ((b v (b' ^ c')) v (b' ^ c))
192, 18letr 137 . . . . 5 (a' ^ c) =< ((b v (b' ^ c')) v (b' ^ c))
201, 19ler2an 173 . . . 4 (a' ^ c) =< ((b' v c) ^ ((b v (b' ^ c')) v (b' ^ c)))
21 leao1 162 . . . . . 6 (a' ^ c') =< (a' v c)
223negantlem8 856 . . . . . 6 (a' v c) = (b' v c)
2321, 22lbtr 139 . . . . 5 (a' ^ c') =< (b' v c)
243negantlem5 853 . . . . . 6 (a' ^ c') = (b' ^ c')
25 leor 159 . . . . . . 7 (b' ^ c') =< (b v (b' ^ c'))
2625ler 149 . . . . . 6 (b' ^ c') =< ((b v (b' ^ c')) v (b' ^ c))
2724, 26bltr 138 . . . . 5 (a' ^ c') =< ((b v (b' ^ c')) v (b' ^ c))
2823, 27ler2an 173 . . . 4 (a' ^ c') =< ((b' v c) ^ ((b v (b' ^ c')) v (b' ^ c)))
2920, 28lel2or 170 . . 3 ((a' ^ c) v (a' ^ c')) =< ((b' v c) ^ ((b v (b' ^ c')) v (b' ^ c)))
30 lear 161 . . . . 5 (a ^ (a' v c)) =< (a' v c)
3130, 22lbtr 139 . . . 4 (a ^ (a' v c)) =< (b' v c)
32 leo 158 . . . . . . 7 a =< (a v (a' ^ c))
3332, 15lbtr 139 . . . . . 6 a =< (b v (b' ^ c))
3433, 17letr 137 . . . . 5 a =< ((b v (b' ^ c')) v (b' ^ c))
3534lel 151 . . . 4 (a ^ (a' v c)) =< ((b v (b' ^ c')) v (b' ^ c))
3631, 35ler2an 173 . . 3 (a ^ (a' v c)) =< ((b' v c) ^ ((b v (b' ^ c')) v (b' ^ c)))
3729, 36lel2or 170 . 2 (((a' ^ c) v (a' ^ c')) v (a ^ (a' v c))) =< ((b' v c) ^ ((b v (b' ^ c')) v (b' ^ c)))
38 df-i3 46 . 2 (a ->3 c) = (((a' ^ c) v (a' ^ c')) v (a ^ (a' v c)))
39 dfi3b 499 . 2 (b ->3 c) = ((b' v c) ^ ((b v (b' ^ c')) v (b' ^ c)))
4037, 38, 39le3tr1 140 1 (a ->3 c) =< (b ->3 c)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->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-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  negant3  860
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