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Theorem lem3.3.6 1056
Description: Equation 3.6 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton, 3-Jul-05.)
Assertion
Ref Expression
lem3.3.6 (a ->2 (b v c)) = ((a v c) ->2 (b v c))
Proof of Theorem lem3.3.6
StepHypRef Expression
1 anor3 90 . . . . . 6 (b' ^ c') = (b v c)'
21ax-r1 35 . . . . 5 (b v c)' = (b' ^ c')
32lan 77 . . . 4 (a' ^ (b v c)') = (a' ^ (b' ^ c'))
4 anandir 115 . . . . 5 ((a' ^ b') ^ c') = ((a' ^ c') ^ (b' ^ c'))
5 anass 76 . . . . 5 ((a' ^ b') ^ c') = (a' ^ (b' ^ c'))
6 anor3 90 . . . . . 6 (a' ^ c') = (a v c)'
76, 12an 79 . . . . 5 ((a' ^ c') ^ (b' ^ c')) = ((a v c)' ^ (b v c)')
84, 5, 73tr2 64 . . . 4 (a' ^ (b' ^ c')) = ((a v c)' ^ (b v c)')
93, 8ax-r2 36 . . 3 (a' ^ (b v c)') = ((a v c)' ^ (b v c)')
109lor 70 . 2 ((b v c) v (a' ^ (b v c)')) = ((b v c) v ((a v c)' ^ (b v c)'))
11 df-i2 45 . 2 (a ->2 (b v c)) = ((b v c) v (a' ^ (b v c)'))
12 df-i2 45 . 2 ((a v c) ->2 (b v c)) = ((b v c) v ((a v c)' ^ (b v c)'))
1310, 11, 123tr1 63 1 (a ->2 (b v c)) = ((a v c) ->2 (b v c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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-t 41  df-f 42  df-i2 45
This theorem is referenced by:  lem3.4.6  1079
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