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Theorem wcom2or 427
 Description: Th. 4.2 Beran p. 49.
Hypotheses
Ref Expression
wfh.1 C (a, b) = 1
wfh.2 C (a, c) = 1
Assertion
Ref Expression
wcom2or C (a, (bc)) = 1

Proof of Theorem wcom2or
StepHypRef Expression
1 wfh.1 . . . . . . . . 9 C (a, b) = 1
21wcomcom 414 . . . . . . . 8 C (b, a) = 1
32wdf-c2 384 . . . . . . 7 (b ≡ ((ba) ∪ (ba ))) = 1
4 ancom 74 . . . . . . . . 9 (ba) = (ab)
5 ancom 74 . . . . . . . . 9 (ba ) = (ab)
64, 52or 72 . . . . . . . 8 ((ba) ∪ (ba )) = ((ab) ∪ (ab))
76bi1 118 . . . . . . 7 (((ba) ∪ (ba )) ≡ ((ab) ∪ (ab))) = 1
83, 7wr2 371 . . . . . 6 (b ≡ ((ab) ∪ (ab))) = 1
9 wfh.2 . . . . . . . . 9 C (a, c) = 1
109wcomcom 414 . . . . . . . 8 C (c, a) = 1
1110wdf-c2 384 . . . . . . 7 (c ≡ ((ca) ∪ (ca ))) = 1
12 ancom 74 . . . . . . . . 9 (ca) = (ac)
13 ancom 74 . . . . . . . . 9 (ca ) = (ac)
1412, 132or 72 . . . . . . . 8 ((ca) ∪ (ca )) = ((ac) ∪ (ac))
1514bi1 118 . . . . . . 7 (((ca) ∪ (ca )) ≡ ((ac) ∪ (ac))) = 1
1611, 15wr2 371 . . . . . 6 (c ≡ ((ac) ∪ (ac))) = 1
178, 16w2or 372 . . . . 5 ((bc) ≡ (((ab) ∪ (ab)) ∪ ((ac) ∪ (ac)))) = 1
18 or4 84 . . . . . 6 (((ab) ∪ (ab)) ∪ ((ac) ∪ (ac))) = (((ab) ∪ (ac)) ∪ ((ab) ∪ (ac)))
1918bi1 118 . . . . 5 ((((ab) ∪ (ab)) ∪ ((ac) ∪ (ac))) ≡ (((ab) ∪ (ac)) ∪ ((ab) ∪ (ac)))) = 1
2017, 19wr2 371 . . . 4 ((bc) ≡ (((ab) ∪ (ac)) ∪ ((ab) ∪ (ac)))) = 1
21 ancom 74 . . . . . . . 8 ((bc) ∩ a) = (a ∩ (bc))
2221bi1 118 . . . . . . 7 (((bc) ∩ a) ≡ (a ∩ (bc))) = 1
231, 9wfh1 423 . . . . . . 7 ((a ∩ (bc)) ≡ ((ab) ∪ (ac))) = 1
2422, 23wr2 371 . . . . . 6 (((bc) ∩ a) ≡ ((ab) ∪ (ac))) = 1
25 ancom 74 . . . . . . . 8 ((bc) ∩ a ) = (a ∩ (bc))
2625bi1 118 . . . . . . 7 (((bc) ∩ a ) ≡ (a ∩ (bc))) = 1
271wcomcom3 416 . . . . . . . 8 C (a , b) = 1
289wcomcom3 416 . . . . . . . 8 C (a , c) = 1
2927, 28wfh1 423 . . . . . . 7 ((a ∩ (bc)) ≡ ((ab) ∪ (ac))) = 1
3026, 29wr2 371 . . . . . 6 (((bc) ∩ a ) ≡ ((ab) ∪ (ac))) = 1
3124, 30w2or 372 . . . . 5 ((((bc) ∩ a) ∪ ((bc) ∩ a )) ≡ (((ab) ∪ (ac)) ∪ ((ab) ∪ (ac)))) = 1
3231wr1 197 . . . 4 ((((ab) ∪ (ac)) ∪ ((ab) ∪ (ac))) ≡ (((bc) ∩ a) ∪ ((bc) ∩ a ))) = 1
3320, 32wr2 371 . . 3 ((bc) ≡ (((bc) ∩ a) ∪ ((bc) ∩ a ))) = 1
3433wdf-c1 383 . 2 C ((bc), a) = 1
3534wcomcom 414 1 C (a, (bc)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29 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  ax-wom 361 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by:  wcom2an  428  ska2  432  ska4  433
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