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Theorem oacom3 1013
Description: Commutation law requiring OA.
Hypotheses
Ref Expression
oacom3.1 (d ^ (a ->2 b)) C ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
oacom3.2 d C (a ->2 b)
Assertion
Ref Expression
oacom3 d C ((a ->2 b) ^ (a ->2 c))
Proof of Theorem oacom3
StepHypRef Expression
1 oacom3.2 . . . . 5 d C (a ->2 b)
21comcom 453 . . . 4 (a ->2 b) C d
3 ancom 74 . . . . . 6 ((a ->2 b) ^ d) = (d ^ (a ->2 b))
4 oacom3.1 . . . . . 6 (d ^ (a ->2 b)) C ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
53, 4bctr 181 . . . . 5 ((a ->2 b) ^ d) C ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
65comcom 453 . . . 4 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) C ((a ->2 b) ^ d)
72, 6gsth2 490 . . 3 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ (a ->2 b)) C d
87comcom 453 . 2 d C (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ (a ->2 b))
9 df-i0 43 . . . 4 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)' v ((a ->2 b) ^ (a ->2 c)))
109ran 78 . . 3 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ (a ->2 b)) = (((b v c)' v ((a ->2 b) ^ (a ->2 c))) ^ (a ->2 b))
11 ancom 74 . . 3 (((b v c)' v ((a ->2 b) ^ (a ->2 c))) ^ (a ->2 b)) = ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
12 oath1 1004 . . 3 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
1310, 11, 123tr 65 . 2 (((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) ^ (a ->2 b)) = ((a ->2 b) ^ (a ->2 c))
148, 13cbtr 182 1 d C ((a ->2 b) ^ (a ->2 c))
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7   ->0 wi0 11   ->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-3oa 998
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  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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