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Theorem oml5 449
Description: Orthomodular law.
Assertion
Ref Expression
oml5 ((a ^ b) v ((a ^ b)' ^ (b v c))) = (b v c)
Proof of Theorem oml5
StepHypRef Expression
1 oml 445 . . 3 ((a ^ b) v ((a ^ b)' ^ ((a ^ b) v (b v c)))) = ((a ^ b) v (b v c))
2 ax-a3 32 . . . . . 6 ((b v (a ^ b)) v c) = (b v ((a ^ b) v c))
3 ancom 74 . . . . . . . . 9 (a ^ b) = (b ^ a)
43lor 70 . . . . . . . 8 (b v (a ^ b)) = (b v (b ^ a))
5 orabs 120 . . . . . . . 8 (b v (b ^ a)) = b
64, 5ax-r2 36 . . . . . . 7 (b v (a ^ b)) = b
76ax-r5 38 . . . . . 6 ((b v (a ^ b)) v c) = (b v c)
8 or12 80 . . . . . 6 (b v ((a ^ b) v c)) = ((a ^ b) v (b v c))
92, 7, 83tr2 64 . . . . 5 (b v c) = ((a ^ b) v (b v c))
109lan 77 . . . 4 ((a ^ b)' ^ (b v c)) = ((a ^ b)' ^ ((a ^ b) v (b v c)))
1110lor 70 . . 3 ((a ^ b) v ((a ^ b)' ^ (b v c))) = ((a ^ b) v ((a ^ b)' ^ ((a ^ b) v (b v c))))
122, 8ax-r2 36 . . 3 ((b v (a ^ b)) v c) = ((a ^ b) v (b v c))
131, 11, 123tr1 63 . 2 ((a ^ b) v ((a ^ b)' ^ (b v c))) = ((b v (a ^ b)) v c)
1413, 7ax-r2 36 1 ((a ^ b) v ((a ^ b)' ^ (b v c))) = (b v 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:  i3th1  543
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