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Theorem com3ii 457
Description: Lemma 3(ii) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
comcom.1 a C b
Assertion
Ref Expression
com3ii (a ^ (a' v b)) = (a ^ b)
Proof of Theorem com3ii
StepHypRef Expression
1 comcom.1 . . . . . 6 a C b
21comcom 453 . . . . 5 b C a
32comd 456 . . . 4 b = ((b v a) ^ (b v a'))
43lan 77 . . 3 (a ^ b) = (a ^ ((b v a) ^ (b v a')))
5 anass 76 . . . . 5 ((a ^ (b v a)) ^ (b v a')) = (a ^ ((b v a) ^ (b v a')))
65ax-r1 35 . . . 4 (a ^ ((b v a) ^ (b v a'))) = ((a ^ (b v a)) ^ (b v a'))
7 ax-a2 31 . . . . . . 7 (b v a) = (a v b)
87lan 77 . . . . . 6 (a ^ (b v a)) = (a ^ (a v b))
9 anabs 121 . . . . . 6 (a ^ (a v b)) = a
108, 9ax-r2 36 . . . . 5 (a ^ (b v a)) = a
11 ax-a2 31 . . . . 5 (b v a') = (a' v b)
1210, 112an 79 . . . 4 ((a ^ (b v a)) ^ (b v a')) = (a ^ (a' v b))
136, 12ax-r2 36 . . 3 (a ^ ((b v a) ^ (b v a'))) = (a ^ (a' v b))
144, 13ax-r2 36 . 2 (a ^ b) = (a ^ (a' v b))
1514ax-r1 35 1 (a ^ (a' v b)) = (a ^ 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-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:  fh1  469  fh2  470
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