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Theorem wwfh4 219
Description: Foulis-Holland Theorem (weak).
Hypotheses
Ref Expression
wwfh4.1 a' C b
wwfh4.2 c C a
Assertion
Ref Expression
wwfh4 ((b v (a ^ c)) == ((b v a) ^ (b v c))) = 1
Proof of Theorem wwfh4
StepHypRef Expression
1 conb 122 . . 3 ((b v (a ^ c)) == ((b v a) ^ (b v c))) = ((b v (a ^ c))' == ((b v a) ^ (b v c))')
2 oran 87 . . . . . 6 (b v (a ^ c)) = (b' ^ (a ^ c)')'
3 df-a 40 . . . . . . . . 9 (a ^ c) = (a' v c')'
43con2 67 . . . . . . . 8 (a ^ c)' = (a' v c')
54lan 77 . . . . . . 7 (b' ^ (a ^ c)') = (b' ^ (a' v c'))
65ax-r4 37 . . . . . 6 (b' ^ (a ^ c)')' = (b' ^ (a' v c'))'
72, 6ax-r2 36 . . . . 5 (b v (a ^ c)) = (b' ^ (a' v c'))'
87con2 67 . . . 4 (b v (a ^ c))' = (b' ^ (a' v c'))
9 df-a 40 . . . . . 6 ((b v a) ^ (b v c)) = ((b v a)' v (b v c)')'
10 oran 87 . . . . . . . . 9 (b v a) = (b' ^ a')'
1110con2 67 . . . . . . . 8 (b v a)' = (b' ^ a')
12 oran 87 . . . . . . . . 9 (b v c) = (b' ^ c')'
1312con2 67 . . . . . . . 8 (b v c)' = (b' ^ c')
1411, 132or 72 . . . . . . 7 ((b v a)' v (b v c)') = ((b' ^ a') v (b' ^ c'))
1514ax-r4 37 . . . . . 6 ((b v a)' v (b v c)')' = ((b' ^ a') v (b' ^ c'))'
169, 15ax-r2 36 . . . . 5 ((b v a) ^ (b v c)) = ((b' ^ a') v (b' ^ c'))'
1716con2 67 . . . 4 ((b v a) ^ (b v c))' = ((b' ^ a') v (b' ^ c'))
188, 172bi 99 . . 3 ((b v (a ^ c))' == ((b v a) ^ (b v c))') = ((b' ^ (a' v c')) == ((b' ^ a') v (b' ^ c')))
191, 18ax-r2 36 . 2 ((b v (a ^ c)) == ((b v a) ^ (b v c))) = ((b' ^ (a' v c')) == ((b' ^ a') v (b' ^ c')))
20 wwfh4.1 . . . 4 a' C b
2120comcom2 183 . . 3 a' C b'
22 ax-a1 30 . . . . . 6 c = c''
2322ax-r1 35 . . . . 5 c'' = c
24 wwfh4.2 . . . . 5 c C a
2523, 24bctr 181 . . . 4 c'' C a
2625comcom2 183 . . 3 c'' C a'
2721, 26wwfh2 217 . 2 ((b' ^ (a' v c')) == ((b' ^ a') v (b' ^ c'))) = 1
2819, 27ax-r2 36 1 ((b v (a ^ c)) == ((b v a) ^ (b 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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