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Theorem elimconslem 867
Description: Lemma for consequent elimination law.
Hypotheses
Ref Expression
elimcons.1 (a ->1 c) = (b ->1 c)
elimcons.2 (a ^ c) =< (b v c')
Assertion
Ref Expression
elimconslem a =< (b v c')
Proof of Theorem elimconslem
StepHypRef Expression
1 df-t 41 . . . . . . 7 1 = ((b v c') v (b v c')')
2 elimcons.2 . . . . . . . . . 10 (a ^ c) =< (b v c')
32lecon 154 . . . . . . . . 9 (b v c')' =< (a ^ c)'
4 oran3 93 . . . . . . . . . 10 (a' v c') = (a ^ c)'
54ax-r1 35 . . . . . . . . 9 (a ^ c)' = (a' v c')
63, 5lbtr 139 . . . . . . . 8 (b v c')' =< (a' v c')
76lelor 166 . . . . . . 7 ((b v c') v (b v c')') =< ((b v c') v (a' v c'))
81, 7bltr 138 . . . . . 6 1 =< ((b v c') v (a' v c'))
98lelan 167 . . . . 5 (a ^ 1) =< (a ^ ((b v c') v (a' v c')))
10 an1 106 . . . . 5 (a ^ 1) = a
11 comor1 461 . . . . . . 7 (a' v c') C a'
1211comcom7 460 . . . . . 6 (a' v c') C a
13 df-a 40 . . . . . . . . . 10 (a ^ c) = (a' v c')'
1413ax-r1 35 . . . . . . . . 9 (a' v c')' = (a ^ c)
1514, 2bltr 138 . . . . . . . 8 (a' v c')' =< (b v c')
1615lecom 180 . . . . . . 7 (a' v c')' C (b v c')
1716comcom6 459 . . . . . 6 (a' v c') C (b v c')
1812, 17fh2c 477 . . . . 5 (a ^ ((b v c') v (a' v c'))) = ((a ^ (b v c')) v (a ^ (a' v c')))
199, 10, 18le3tr2 141 . . . 4 a =< ((a ^ (b v c')) v (a ^ (a' v c')))
20 elimcons.1 . . . . . . . . 9 (a ->1 c) = (b ->1 c)
21 df-i1 44 . . . . . . . . 9 (a ->1 c) = (a' v (a ^ c))
22 df-i1 44 . . . . . . . . 9 (b ->1 c) = (b' v (b ^ c))
2320, 21, 223tr2 64 . . . . . . . 8 (a' v (a ^ c)) = (b' v (b ^ c))
2413lor 70 . . . . . . . 8 (a' v (a ^ c)) = (a' v (a' v c')')
25 df-a 40 . . . . . . . . 9 (b ^ c) = (b' v c')'
2625lor 70 . . . . . . . 8 (b' v (b ^ c)) = (b' v (b' v c')')
2723, 24, 263tr2 64 . . . . . . 7 (a' v (a' v c')') = (b' v (b' v c')')
2827ax-r4 37 . . . . . 6 (a' v (a' v c')')' = (b' v (b' v c')')'
29 df-a 40 . . . . . 6 (a ^ (a' v c')) = (a' v (a' v c')')'
30 df-a 40 . . . . . 6 (b ^ (b' v c')) = (b' v (b' v c')')'
3128, 29, 303tr1 63 . . . . 5 (a ^ (a' v c')) = (b ^ (b' v c'))
3231lor 70 . . . 4 ((a ^ (b v c')) v (a ^ (a' v c'))) = ((a ^ (b v c')) v (b ^ (b' v c')))
3319, 32lbtr 139 . . 3 a =< ((a ^ (b v c')) v (b ^ (b' v c')))
34 lear 161 . . . 4 (a ^ (b v c')) =< (b v c')
3534leror 152 . . 3 ((a ^ (b v c')) v (b ^ (b' v c'))) =< ((b v c') v (b ^ (b' v c')))
3633, 35letr 137 . 2 a =< ((b v c') v (b ^ (b' v c')))
37 ax-a2 31 . . 3 ((b v c') v (b ^ (b' v c'))) = ((b ^ (b' v c')) v (b v c'))
38 leao1 162 . . . 4 (b ^ (b' v c')) =< (b v c')
3938df-le2 131 . . 3 ((b ^ (b' v c')) v (b v c')) = (b v c')
4037, 39ax-r2 36 . 2 ((b v c') v (b ^ (b' v c'))) = (b v c')
4136, 40lbtr 139 1 a =< (b v c')
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:  elimcons  868
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