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Theorem mi 125
Description: Mittelstaedt implication.
Assertion
Ref Expression
mi ((a v b) == b) = (b v (a' ^ b'))
Proof of Theorem mi
StepHypRef Expression
1 dfb 94 . 2 ((a v b) == b) = (((a v b) ^ b) v ((a v b)' ^ b'))
2 ancom 74 . . . 4 ((a v b) ^ b) = (b ^ (a v b))
3 ax-a2 31 . . . . . 6 (a v b) = (b v a)
43lan 77 . . . . 5 (b ^ (a v b)) = (b ^ (b v a))
5 anabs 121 . . . . 5 (b ^ (b v a)) = b
64, 5ax-r2 36 . . . 4 (b ^ (a v b)) = b
72, 6ax-r2 36 . . 3 ((a v b) ^ b) = b
8 oran 87 . . . . . . 7 (a v b) = (a' ^ b')'
98con2 67 . . . . . 6 (a v b)' = (a' ^ b')
109ran 78 . . . . 5 ((a v b)' ^ b') = ((a' ^ b') ^ b')
11 anass 76 . . . . 5 ((a' ^ b') ^ b') = (a' ^ (b' ^ b'))
1210, 11ax-r2 36 . . . 4 ((a v b)' ^ b') = (a' ^ (b' ^ b'))
13 anidm 111 . . . . 5 (b' ^ b') = b'
1413lan 77 . . . 4 (a' ^ (b' ^ b')) = (a' ^ b')
1512, 14ax-r2 36 . . 3 ((a v b)' ^ b') = (a' ^ b')
167, 152or 72 . 2 (((a v b) ^ b) v ((a v b)' ^ b')) = (b v (a' ^ b'))
171, 16ax-r2 36 1 ((a v b) == b) = (b v (a' ^ b'))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42
This theorem is referenced by:  di  126  lei2  346
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