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Theorem 2oath1 826
Description: OA-like theorem with ->2 instead of ->0.
Assertion
Ref Expression
2oath1 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
Proof of Theorem 2oath1
StepHypRef Expression
1 df-i2 45 . . 3 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))
21lan 77 . 2 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')))
3 coman1 185 . . 3 ((a ->2 b) ^ (a ->2 c)) C (a ->2 b)
4 comorr2 463 . . . . 5 ((a ->2 b) ^ (a ->2 c)) C ((b v c) v ((a ->2 b) ^ (a ->2 c)))
54comcom2 183 . . . 4 ((a ->2 b) ^ (a ->2 c)) C ((b v c) v ((a ->2 b) ^ (a ->2 c)))'
6 anor3 90 . . . . 5 ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))') = ((b v c) v ((a ->2 b) ^ (a ->2 c)))'
76ax-r1 35 . . . 4 ((b v c) v ((a ->2 b) ^ (a ->2 c)))' = ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')
85, 7cbtr 182 . . 3 ((a ->2 b) ^ (a ->2 c)) C ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')
93, 8fh2 470 . 2 ((a ->2 b) ^ (((a ->2 b) ^ (a ->2 c)) v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))) = (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')))
10 anass 76 . . . . . 6 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
1110ax-r1 35 . . . . 5 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c))
12 anidm 111 . . . . . 6 ((a ->2 b) ^ (a ->2 b)) = (a ->2 b)
1312ran 78 . . . . 5 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
1411, 13ax-r2 36 . . . 4 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ (a ->2 c))
15 oran 87 . . . . . . . . 9 ((b v c) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')'
1615lor 70 . . . . . . . 8 ((a ->2 b)' v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b)' v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')')
1716ax-r1 35 . . . . . . 7 ((a ->2 b)' v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')') = ((a ->2 b)' v ((b v c) v ((a ->2 b) ^ (a ->2 c))))
18 2oalem1 825 . . . . . . 7 ((a ->2 b)' v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = 1
1917, 18ax-r2 36 . . . . . 6 ((a ->2 b)' v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')') = 1
2019ax-r4 37 . . . . 5 ((a ->2 b)' v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')')' = 1'
21 df-a 40 . . . . 5 ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')) = ((a ->2 b)' v ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')')'
22 df-f 42 . . . . 5 0 = 1'
2320, 21, 223tr1 63 . . . 4 ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))')) = 0
2414, 232or 72 . . 3 (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))) = (((a ->2 b) ^ (a ->2 c)) v 0)
25 or0 102 . . 3 (((a ->2 b) ^ (a ->2 c)) v 0) = ((a ->2 b) ^ (a ->2 c))
2624, 25ax-r2 36 . 2 (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)' ^ ((a ->2 b) ^ (a ->2 c))'))) = ((a ->2 b) ^ (a ->2 c))
272, 9, 263tr 65 1 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->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
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:  2oath1i1  827  oale  829  oaeqv  830  distoa  944
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