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Theorem wwfh2 217
Description: Foulis-Holland Theorem (weak).
Hypotheses
Ref Expression
wwfh2.1 a C b
wwfh2.2 c' C a
Assertion
Ref Expression
wwfh2 ((b ^ (a v c)) == ((b ^ a) v (b ^ c))) = 1
Proof of Theorem wwfh2
StepHypRef Expression
1 bicom 96 . 2 ((b ^ (a v c)) == ((b ^ a) v (b ^ c))) = (((b ^ a) v (b ^ c)) == (b ^ (a v c)))
2 ledi 174 . . 3 ((b ^ a) v (b ^ c)) =< (b ^ (a v c))
3 oran 87 . . . . . . . . . . 11 ((b ^ a) v (b ^ c)) = ((b ^ a)' ^ (b ^ c)')'
4 df-a 40 . . . . . . . . . . . . . 14 (b ^ a) = (b' v a')'
54con2 67 . . . . . . . . . . . . 13 (b ^ a)' = (b' v a')
65ran 78 . . . . . . . . . . . 12 ((b ^ a)' ^ (b ^ c)') = ((b' v a') ^ (b ^ c)')
76ax-r4 37 . . . . . . . . . . 11 ((b ^ a)' ^ (b ^ c)')' = ((b' v a') ^ (b ^ c)')'
83, 7ax-r2 36 . . . . . . . . . 10 ((b ^ a) v (b ^ c)) = ((b' v a') ^ (b ^ c)')'
98con2 67 . . . . . . . . 9 ((b ^ a) v (b ^ c))' = ((b' v a') ^ (b ^ c)')
109lan 77 . . . . . . . 8 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((b ^ (a v c)) ^ ((b' v a') ^ (b ^ c)'))
11 an4 86 . . . . . . . . 9 ((b ^ (a v c)) ^ ((b' v a') ^ (b ^ c)')) = ((b ^ (b' v a')) ^ ((a v c) ^ (b ^ c)'))
12 ax-a1 30 . . . . . . . . . . . . . 14 a = a''
1312ax-r1 35 . . . . . . . . . . . . 13 a'' = a
14 wwfh2.1 . . . . . . . . . . . . 13 a C b
1513, 14bctr 181 . . . . . . . . . . . 12 a'' C b
1615wwcom3ii 215 . . . . . . . . . . 11 (b ^ (b' v a')) = (b ^ a')
17 ancom 74 . . . . . . . . . . 11 (b ^ a') = (a' ^ b)
1816, 17ax-r2 36 . . . . . . . . . 10 (b ^ (b' v a')) = (a' ^ b)
1918ran 78 . . . . . . . . 9 ((b ^ (b' v a')) ^ ((a v c) ^ (b ^ c)')) = ((a' ^ b) ^ ((a v c) ^ (b ^ c)'))
2011, 19ax-r2 36 . . . . . . . 8 ((b ^ (a v c)) ^ ((b' v a') ^ (b ^ c)')) = ((a' ^ b) ^ ((a v c) ^ (b ^ c)'))
2110, 20ax-r2 36 . . . . . . 7 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((a' ^ b) ^ ((a v c) ^ (b ^ c)'))
22 an4 86 . . . . . . 7 ((a' ^ b) ^ ((a v c) ^ (b ^ c)')) = ((a' ^ (a v c)) ^ (b ^ (b ^ c)'))
2321, 22ax-r2 36 . . . . . 6 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((a' ^ (a v c)) ^ (b ^ (b ^ c)'))
2412ax-r5 38 . . . . . . . . 9 (a v c) = (a'' v c)
2524lan 77 . . . . . . . 8 (a' ^ (a v c)) = (a' ^ (a'' v c))
26 wwfh2.2 . . . . . . . . . 10 c' C a
2726comcom2 183 . . . . . . . . 9 c' C a'
2827wwcom3ii 215 . . . . . . . 8 (a' ^ (a'' v c)) = (a' ^ c)
2925, 28ax-r2 36 . . . . . . 7 (a' ^ (a v c)) = (a' ^ c)
3029ran 78 . . . . . 6 ((a' ^ (a v c)) ^ (b ^ (b ^ c)')) = ((a' ^ c) ^ (b ^ (b ^ c)'))
3123, 30ax-r2 36 . . . . 5 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((a' ^ c) ^ (b ^ (b ^ c)'))
32 anass 76 . . . . 5 ((a' ^ c) ^ (b ^ (b ^ c)')) = (a' ^ (c ^ (b ^ (b ^ c)')))
3331, 32ax-r2 36 . . . 4 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = (a' ^ (c ^ (b ^ (b ^ c)')))
34 anass 76 . . . . . . . 8 ((b ^ c) ^ (b ^ c)') = (b ^ (c ^ (b ^ c)'))
3534ax-r1 35 . . . . . . 7 (b ^ (c ^ (b ^ c)')) = ((b ^ c) ^ (b ^ c)')
36 an12 81 . . . . . . 7 (c ^ (b ^ (b ^ c)')) = (b ^ (c ^ (b ^ c)'))
37 dff 101 . . . . . . 7 0 = ((b ^ c) ^ (b ^ c)')
3835, 36, 373tr1 63 . . . . . 6 (c ^ (b ^ (b ^ c)')) = 0
3938lan 77 . . . . 5 (a' ^ (c ^ (b ^ (b ^ c)'))) = (a' ^ 0)
40 an0 108 . . . . 5 (a' ^ 0) = 0
4139, 40ax-r2 36 . . . 4 (a' ^ (c ^ (b ^ (b ^ c)'))) = 0
4233, 41ax-r2 36 . . 3 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = 0
432, 42wwoml3 213 . 2 (((b ^ a) v (b ^ c)) == (b ^ (a v c))) = 1
441, 43ax-r2 36 1 ((b ^ (a v c)) == ((b ^ a) v (b ^ c))) = 1
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8  0wf 9
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:  wwfh4  219
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