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Theorem oml5a 450
Description: Orthomodular law.
Assertion
Ref Expression
oml5a ((a v b) ^ ((a v b)' v (b ^ c))) = (b ^ c)
Proof of Theorem oml5a
StepHypRef Expression
1 omla 447 . . 3 ((a v b) ^ ((a v b)' v ((a v b) ^ (b ^ c)))) = ((a v b) ^ (b ^ c))
2 anass 76 . . . . . 6 ((b ^ (a v b)) ^ c) = (b ^ ((a v b) ^ c))
3 ax-a2 31 . . . . . . . . 9 (a v b) = (b v a)
43lan 77 . . . . . . . 8 (b ^ (a v b)) = (b ^ (b v a))
5 anabs 121 . . . . . . . 8 (b ^ (b v a)) = b
64, 5ax-r2 36 . . . . . . 7 (b ^ (a v b)) = b
76ran 78 . . . . . 6 ((b ^ (a v b)) ^ c) = (b ^ c)
8 an12 81 . . . . . 6 (b ^ ((a v b) ^ c)) = ((a v b) ^ (b ^ c))
92, 7, 83tr2 64 . . . . 5 (b ^ c) = ((a v b) ^ (b ^ c))
109lor 70 . . . 4 ((a v b)' v (b ^ c)) = ((a v b)' v ((a v b) ^ (b ^ c)))
1110lan 77 . . 3 ((a v b) ^ ((a v b)' v (b ^ c))) = ((a v b) ^ ((a v b)' v ((a v b) ^ (b ^ c))))
122, 8ax-r2 36 . . 3 ((b ^ (a v b)) ^ c) = ((a v b) ^ (b ^ c))
131, 11, 123tr1 63 . 2 ((a v b) ^ ((a v b)' v (b ^ c))) = ((b ^ (a v b)) ^ c)
1413, 7ax-r2 36 1 ((a v b) ^ ((a v b)' v (b ^ c))) = (b ^ c)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7
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  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42
This theorem is referenced by: (None)
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