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Theorem womle2a 295
Description: An equivalent to the WOM law.
Hypothesis
Ref Expression
womle2a.1 (a ^ (a ->2 b)) =< ((a ->2 b)' v (a ->1 b))
Assertion
Ref Expression
womle2a ((a ->2 b)' v (a ->1 b)) = 1
Proof of Theorem womle2a
StepHypRef Expression
1 or4 84 . . 3 (((a ->2 b)' v (a ->2 b)') v ((a ->1 b) v a')) = (((a ->2 b)' v (a ->1 b)) v ((a ->2 b)' v a'))
2 oridm 110 . . . 4 ((a ->2 b)' v (a ->2 b)') = (a ->2 b)'
3 df-i1 44 . . . . . 6 (a ->1 b) = (a' v (a ^ b))
43ax-r5 38 . . . . 5 ((a ->1 b) v a') = ((a' v (a ^ b)) v a')
5 oridm 110 . . . . . . 7 (a' v a') = a'
65ax-r5 38 . . . . . 6 ((a' v a') v (a ^ b)) = (a' v (a ^ b))
7 or32 82 . . . . . 6 ((a' v (a ^ b)) v a') = ((a' v a') v (a ^ b))
86, 7, 33tr1 63 . . . . 5 ((a' v (a ^ b)) v a') = (a ->1 b)
94, 8ax-r2 36 . . . 4 ((a ->1 b) v a') = (a ->1 b)
102, 92or 72 . . 3 (((a ->2 b)' v (a ->2 b)') v ((a ->1 b) v a')) = ((a ->2 b)' v (a ->1 b))
11 ax-a2 31 . . . . 5 ((a ->2 b)' v a') = (a' v (a ->2 b)')
12 oran3 93 . . . . 5 (a' v (a ->2 b)') = (a ^ (a ->2 b))'
1311, 12ax-r2 36 . . . 4 ((a ->2 b)' v a') = (a ^ (a ->2 b))'
1413lor 70 . . 3 (((a ->2 b)' v (a ->1 b)) v ((a ->2 b)' v a')) = (((a ->2 b)' v (a ->1 b)) v (a ^ (a ->2 b))')
151, 10, 143tr2 64 . 2 ((a ->2 b)' v (a ->1 b)) = (((a ->2 b)' v (a ->1 b)) v (a ^ (a ->2 b))')
16 le1 146 . . 3 (((a ->2 b)' v (a ->1 b)) v (a ^ (a ->2 b))') =< 1
17 df-t 41 . . . 4 1 = ((a ^ (a ->2 b)) v (a ^ (a ->2 b))')
18 womle2a.1 . . . . 5 (a ^ (a ->2 b)) =< ((a ->2 b)' v (a ->1 b))
1918leror 152 . . . 4 ((a ^ (a ->2 b)) v (a ^ (a ->2 b))') =< (((a ->2 b)' v (a ->1 b)) v (a ^ (a ->2 b))')
2017, 19bltr 138 . . 3 1 =< (((a ->2 b)' v (a ->1 b)) v (a ^ (a ->2 b))')
2116, 20lebi 145 . 2 (((a ->2 b)' v (a ->1 b)) v (a ^ (a ->2 b))') = 1
2215, 21ax-r2 36 1 ((a ->2 b)' v (a ->1 b)) = 1
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12   ->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
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131
This theorem is referenced by:  womle  298
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