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Theorem i1abs 801
Description: An absorption law for ->1.
Assertion
Ref Expression
i1abs ((a ->1 b)' v (a ^ b)) = a
Proof of Theorem i1abs
StepHypRef Expression
1 ud1lem0c 277 . . 3 (a ->1 b)' = (a ^ (a' v b'))
21ax-r5 38 . 2 ((a ->1 b)' v (a ^ b)) = ((a ^ (a' v b')) v (a ^ b))
3 comanr1 464 . . 3 a C (a ^ b)
4 comorr 184 . . . 4 a' C (a' v b')
54comcom6 459 . . 3 a C (a' v b')
63, 5fh4r 476 . 2 ((a ^ (a' v b')) v (a ^ b)) = ((a v (a ^ b)) ^ ((a' v b') v (a ^ b)))
7 orabs 120 . . . 4 (a v (a ^ b)) = a
8 df-a 40 . . . . . 6 (a ^ b) = (a' v b')'
98lor 70 . . . . 5 ((a' v b') v (a ^ b)) = ((a' v b') v (a' v b')')
10 df-t 41 . . . . . 6 1 = ((a' v b') v (a' v b')')
1110ax-r1 35 . . . . 5 ((a' v b') v (a' v b')') = 1
129, 11ax-r2 36 . . . 4 ((a' v b') v (a ^ b)) = 1
137, 122an 79 . . 3 ((a v (a ^ b)) ^ ((a' v b') v (a ^ b))) = (a ^ 1)
14 an1 106 . . 3 (a ^ 1) = a
1513, 14ax-r2 36 . 2 ((a v (a ^ b)) ^ ((a' v b') v (a ^ b))) = a
162, 6, 153tr 65 1 ((a ->1 b)' v (a ^ b)) = a
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  cancellem  891
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