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Theorem oa43v 1028
Description: Derivation of 3-variable OA from 4-variable OA.
Assertion
Ref Expression
oa43v ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
Proof of Theorem oa43v
StepHypRef Expression
1 ud2lem0c 278 . . . . 5 (a ->2 c)' = (c' ^ (a v c))
2 lea 160 . . . . 5 (c' ^ (a v c)) =< c'
31, 2bltr 138 . . . 4 (a ->2 c)' =< c'
4 ud2lem0c 278 . . . . 5 (a ->2 b)' = (b' ^ (a v b))
5 lea 160 . . . . 5 (b' ^ (a v b)) =< b'
64, 5bltr 138 . . . 4 (a ->2 b)' =< b'
73, 6ax-oal4 1026 . . . 4 (((a ->2 c)' v c) ^ ((a ->2 b)' v b)) =< (c v ((a ->2 c)' ^ ((a ->2 b)' v (((a ->2 c)' v (a ->2 b)') ^ (c v b)))))
8 id 59 . . . 4 (a ->2 c)' = (a ->2 c)'
9 id 59 . . . 4 (a ->2 b)' = (a ->2 b)'
103, 6, 7, 8, 9oa4v3v 934 . . 3 (c' ^ ((a ->2 c) v ((a ->2 b) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))))) =< ((c' ^ (a ->2 c)) v (b' ^ (a ->2 b)))
1110oal42 935 . 2 (c' ^ ((a ->2 c) v ((a ->2 b) ^ ((c v b)' v ((a ->2 c) ^ (a ->2 b)))))) =< a'
1211oa23 936 1 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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  ax-oal4 1026
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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