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Theorem u12lem 771
Description: Implication lemma.
Assertion
Ref Expression
u12lem ((a ->1 b) v (a ->2 b)) = (a ->0 b)
Proof of Theorem u12lem
StepHypRef Expression
1 orordi 112 . . 3 ((a ->1 b) v (b v (a' ^ b'))) = (((a ->1 b) v b) v ((a ->1 b) v (a' ^ b')))
2 u1lemob 630 . . . . 5 ((a ->1 b) v b) = (a' v b)
3 df-i1 44 . . . . . . 7 (a ->1 b) = (a' v (a ^ b))
43ax-r5 38 . . . . . 6 ((a ->1 b) v (a' ^ b')) = ((a' v (a ^ b)) v (a' ^ b'))
5 or32 82 . . . . . . 7 ((a' v (a ^ b)) v (a' ^ b')) = ((a' v (a' ^ b')) v (a ^ b))
6 orabs 120 . . . . . . . 8 (a' v (a' ^ b')) = a'
76ax-r5 38 . . . . . . 7 ((a' v (a' ^ b')) v (a ^ b)) = (a' v (a ^ b))
85, 7ax-r2 36 . . . . . 6 ((a' v (a ^ b)) v (a' ^ b')) = (a' v (a ^ b))
94, 8ax-r2 36 . . . . 5 ((a ->1 b) v (a' ^ b')) = (a' v (a ^ b))
102, 92or 72 . . . 4 (((a ->1 b) v b) v ((a ->1 b) v (a' ^ b'))) = ((a' v b) v (a' v (a ^ b)))
11 id 59 . . . . . . 7 (a' v b) = (a' v b)
1211bile 142 . . . . . 6 (a' v b) =< (a' v b)
13 lear 161 . . . . . . 7 (a ^ b) =< b
1413lelor 166 . . . . . 6 (a' v (a ^ b)) =< (a' v b)
1512, 14lel2or 170 . . . . 5 ((a' v b) v (a' v (a ^ b))) =< (a' v b)
16 leo 158 . . . . 5 (a' v b) =< ((a' v b) v (a' v (a ^ b)))
1715, 16lebi 145 . . . 4 ((a' v b) v (a' v (a ^ b))) = (a' v b)
1810, 17ax-r2 36 . . 3 (((a ->1 b) v b) v ((a ->1 b) v (a' ^ b'))) = (a' v b)
191, 18ax-r2 36 . 2 ((a ->1 b) v (b v (a' ^ b'))) = (a' v b)
20 df-i2 45 . . 3 (a ->2 b) = (b v (a' ^ b'))
2120lor 70 . 2 ((a ->1 b) v (a ->2 b)) = ((a ->1 b) v (b v (a' ^ b')))
22 df-i0 43 . 2 (a ->0 b) = (a' v b)
2319, 21, 223tr1 63 1 ((a ->1 b) v (a ->2 b)) = (a ->0 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->0 wi0 11   ->1 wi1 12   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  distoah2  941  distoah3  942  distoa  944  d3oa  995  oadist2b  1008  oadist12  1010  lem4.6.6i1j2  1091
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