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Theorem com3iia 1100
Description: The dual of com3ii 457. (Contributed by Roy F. Longton, 3-Jul-05.)
Hypothesis
Ref Expression
com3iia.1 a C b
Assertion
Ref Expression
com3iia (a v (a' ^ b)) = (a v b)
Proof of Theorem com3iia
StepHypRef Expression
1 comid 187 . . . 4 a C a
21comcom2 183 . . 3 a C a'
3 com3iia.1 . . 3 a C b
42, 3fh3 471 . 2 (a v (a' ^ b)) = ((a v a') ^ (a v b))
5 lear 161 . . 3 ((a v a') ^ (a v b)) =< (a v b)
6 ax-a4 33 . . . . 5 ((a v b) v (a v a')) = (a v a')
76df-le1 130 . . . 4 (a v b) =< (a v a')
8 leid 148 . . . 4 (a v b) =< (a v b)
97, 8ler2an 173 . . 3 (a v b) =< ((a v a') ^ (a v b))
105, 9lebi 145 . 2 ((a v a') ^ (a v b)) = (a v b)
114, 10ax-r2 36 1 (a v (a' ^ b)) = (a v b)
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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