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Theorem wwcom3ii 215
Description: Lemma 3(ii) (weak) of Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wwcom3ii.1 b' C a
Assertion
Ref Expression
wwcom3ii (a ^ (a' v b)) = (a ^ b)
Proof of Theorem wwcom3ii
StepHypRef Expression
1 wwcom3ii.1 . . . . 5 b' C a
21wwcomd 214 . . . 4 b = ((b v a) ^ (b v a'))
32lan 77 . . 3 (a ^ b) = (a ^ ((b v a) ^ (b v a')))
4 anass 76 . . . . 5 ((a ^ (b v a)) ^ (b v a')) = (a ^ ((b v a) ^ (b v a')))
54ax-r1 35 . . . 4 (a ^ ((b v a) ^ (b v a'))) = ((a ^ (b v a)) ^ (b v a'))
6 ax-a2 31 . . . . . . 7 (b v a) = (a v b)
76lan 77 . . . . . 6 (a ^ (b v a)) = (a ^ (a v b))
8 anabs 121 . . . . . 6 (a ^ (a v b)) = a
97, 8ax-r2 36 . . . . 5 (a ^ (b v a)) = a
10 ax-a2 31 . . . . 5 (b v a') = (a' v b)
119, 102an 79 . . . 4 ((a ^ (b v a)) ^ (b v a')) = (a ^ (a' v b))
125, 11ax-r2 36 . . 3 (a ^ ((b v a) ^ (b v a'))) = (a ^ (a' v b))
133, 12ax-r2 36 . 2 (a ^ b) = (a ^ (a' v b))
1413ax-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
This theorem depends on definitions:  df-a 40  df-c2 133
This theorem is referenced by:  wwfh1  216  wwfh2  217
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