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Theorem elimcons2 869
Description: Consequent elimination law.
Hypotheses
Ref Expression
elimcons2.1 (a ->1 c) = (b ->1 c)
elimcons2.2 (a ^ (c ^ (b ->1 c))) =< (b v (c' v (a ->1 c)'))
Assertion
Ref Expression
elimcons2 a =< b
Proof of Theorem elimcons2
StepHypRef Expression
1 elimcons2.1 . 2 (a ->1 c) = (b ->1 c)
2 elimcons2.2 . . 3 (a ^ (c ^ (b ->1 c))) =< (b v (c' v (a ->1 c)'))
31ax-r1 35 . . . . . . 7 (b ->1 c) = (a ->1 c)
4 df-i1 44 . . . . . . 7 (a ->1 c) = (a' v (a ^ c))
53, 4ax-r2 36 . . . . . 6 (b ->1 c) = (a' v (a ^ c))
65lan 77 . . . . 5 (c ^ (b ->1 c)) = (c ^ (a' v (a ^ c)))
76lan 77 . . . 4 (a ^ (c ^ (b ->1 c))) = (a ^ (c ^ (a' v (a ^ c))))
8 anass 76 . . . . 5 ((a ^ c) ^ (a' v (a ^ c))) = (a ^ (c ^ (a' v (a ^ c))))
98ax-r1 35 . . . 4 (a ^ (c ^ (a' v (a ^ c)))) = ((a ^ c) ^ (a' v (a ^ c)))
10 leor 159 . . . . 5 (a ^ c) =< (a' v (a ^ c))
1110df2le2 136 . . . 4 ((a ^ c) ^ (a' v (a ^ c))) = (a ^ c)
127, 9, 113tr 65 . . 3 (a ^ (c ^ (b ->1 c))) = (a ^ c)
131ax-r4 37 . . . . . . . 8 (a ->1 c)' = (b ->1 c)'
14 ud1lem0c 277 . . . . . . . 8 (b ->1 c)' = (b ^ (b' v c'))
1513, 14ax-r2 36 . . . . . . 7 (a ->1 c)' = (b ^ (b' v c'))
1615lor 70 . . . . . 6 (c' v (a ->1 c)') = (c' v (b ^ (b' v c')))
17 ax-a2 31 . . . . . 6 (c' v (b ^ (b' v c'))) = ((b ^ (b' v c')) v c')
1816, 17ax-r2 36 . . . . 5 (c' v (a ->1 c)') = ((b ^ (b' v c')) v c')
1918lor 70 . . . 4 (b v (c' v (a ->1 c)')) = (b v ((b ^ (b' v c')) v c'))
20 ax-a3 32 . . . . 5 ((b v (b ^ (b' v c'))) v c') = (b v ((b ^ (b' v c')) v c'))
2120ax-r1 35 . . . 4 (b v ((b ^ (b' v c')) v c')) = ((b v (b ^ (b' v c'))) v c')
22 ax-a2 31 . . . . . 6 (b v (b ^ (b' v c'))) = ((b ^ (b' v c')) v b)
23 lea 160 . . . . . . 7 (b ^ (b' v c')) =< b
2423df-le2 131 . . . . . 6 ((b ^ (b' v c')) v b) = b
2522, 24ax-r2 36 . . . . 5 (b v (b ^ (b' v c'))) = b
2625ax-r5 38 . . . 4 ((b v (b ^ (b' v c'))) v c') = (b v c')
2719, 21, 263tr 65 . . 3 (b v (c' v (a ->1 c)')) = (b v c')
282, 12, 27le3tr2 141 . 2 (a ^ c) =< (b v c')
291, 28elimcons 868 1 a =< b
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->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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