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Theorem wwfh3 218
Description: Foulis-Holland Theorem (weak).
Hypotheses
Ref Expression
wwfh3.1 b' C a
wwfh3.2 c' C a
Assertion
Ref Expression
wwfh3 ((a v (b ^ c)) == ((a v b) ^ (a v c))) = 1
Proof of Theorem wwfh3
StepHypRef Expression
1 conb 122 . . 3 ((a v (b ^ c)) == ((a v b) ^ (a v c))) = ((a v (b ^ c))' == ((a v b) ^ (a v c))')
2 oran 87 . . . . . 6 (a v (b ^ c)) = (a' ^ (b ^ c)')'
3 df-a 40 . . . . . . . . 9 (b ^ c) = (b' v c')'
43con2 67 . . . . . . . 8 (b ^ c)' = (b' v c')
54lan 77 . . . . . . 7 (a' ^ (b ^ c)') = (a' ^ (b' v c'))
65ax-r4 37 . . . . . 6 (a' ^ (b ^ c)')' = (a' ^ (b' v c'))'
72, 6ax-r2 36 . . . . 5 (a v (b ^ c)) = (a' ^ (b' v c'))'
87con2 67 . . . 4 (a v (b ^ c))' = (a' ^ (b' v c'))
9 df-a 40 . . . . . 6 ((a v b) ^ (a v c)) = ((a v b)' v (a v c)')'
10 oran 87 . . . . . . . . 9 (a v b) = (a' ^ b')'
1110con2 67 . . . . . . . 8 (a v b)' = (a' ^ b')
12 oran 87 . . . . . . . . 9 (a v c) = (a' ^ c')'
1312con2 67 . . . . . . . 8 (a v c)' = (a' ^ c')
1411, 132or 72 . . . . . . 7 ((a v b)' v (a v c)') = ((a' ^ b') v (a' ^ c'))
1514ax-r4 37 . . . . . 6 ((a v b)' v (a v c)')' = ((a' ^ b') v (a' ^ c'))'
169, 15ax-r2 36 . . . . 5 ((a v b) ^ (a v c)) = ((a' ^ b') v (a' ^ c'))'
1716con2 67 . . . 4 ((a v b) ^ (a v c))' = ((a' ^ b') v (a' ^ c'))
188, 172bi 99 . . 3 ((a v (b ^ c))' == ((a v b) ^ (a v c))') = ((a' ^ (b' v c')) == ((a' ^ b') v (a' ^ c')))
191, 18ax-r2 36 . 2 ((a v (b ^ c)) == ((a v b) ^ (a v c))) = ((a' ^ (b' v c')) == ((a' ^ b') v (a' ^ c')))
20 wwfh3.1 . . . 4 b' C a
2120comcom2 183 . . 3 b' C a'
22 wwfh3.2 . . . 4 c' C a
2322comcom2 183 . . 3 c' C a'
2421, 23wwfh1 216 . 2 ((a' ^ (b' v c')) == ((a' ^ b') v (a' ^ c'))) = 1
2519, 24ax-r2 36 1 ((a v (b ^ c)) == ((a v b) ^ (a v c))) = 1
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
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
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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