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Theorem wfh1 423
 Description: Weak structural analog of Foulis-Holland Theorem.
Hypotheses
Ref Expression
wfh.1 C (a, b) = 1
wfh.2 C (a, c) = 1
Assertion
Ref Expression
wfh1 ((a ∩ (bc)) ≡ ((ab) ∪ (ac))) = 1

Proof of Theorem wfh1
StepHypRef Expression
1 wledi 405 . . 3 (((ab) ∪ (ac)) ≤2 (a ∩ (bc))) = 1
2 ancom 74 . . . . . . 7 (a ∩ (bc)) = ((bc) ∩ a)
32bi1 118 . . . . . 6 ((a ∩ (bc)) ≡ ((bc) ∩ a)) = 1
4 df-a 40 . . . . . . . . . 10 (ab) = (ab )
54bi1 118 . . . . . . . . 9 ((ab) ≡ (ab ) ) = 1
6 df-a 40 . . . . . . . . . 10 (ac) = (ac )
76bi1 118 . . . . . . . . 9 ((ac) ≡ (ac ) ) = 1
85, 7w2or 372 . . . . . . . 8 (((ab) ∪ (ac)) ≡ ((ab ) ∪ (ac ) )) = 1
9 df-a 40 . . . . . . . . . . 11 ((ab ) ∩ (ac )) = ((ab ) ∪ (ac ) )
109bi1 118 . . . . . . . . . 10 (((ab ) ∩ (ac )) ≡ ((ab ) ∪ (ac ) ) ) = 1
1110wr1 197 . . . . . . . . 9 (((ab ) ∪ (ac ) ) ≡ ((ab ) ∩ (ac ))) = 1
1211wcon3 209 . . . . . . . 8 (((ab ) ∪ (ac ) ) ≡ ((ab ) ∩ (ac )) ) = 1
138, 12wr2 371 . . . . . . 7 (((ab) ∪ (ac)) ≡ ((ab ) ∩ (ac )) ) = 1
1413wcon2 208 . . . . . 6 (((ab) ∪ (ac)) ≡ ((ab ) ∩ (ac ))) = 1
153, 14w2an 373 . . . . 5 (((a ∩ (bc)) ∩ ((ab) ∪ (ac)) ) ≡ (((bc) ∩ a) ∩ ((ab ) ∩ (ac )))) = 1
16 anass 76 . . . . . . . 8 (((bc) ∩ a) ∩ ((ab ) ∩ (ac ))) = ((bc) ∩ (a ∩ ((ab ) ∩ (ac ))))
1716bi1 118 . . . . . . 7 ((((bc) ∩ a) ∩ ((ab ) ∩ (ac ))) ≡ ((bc) ∩ (a ∩ ((ab ) ∩ (ac ))))) = 1
18 wfh.1 . . . . . . . . . . . 12 C (a, b) = 1
1918wcomcom2 415 . . . . . . . . . . 11 C (a, b ) = 1
2019wcom3ii 419 . . . . . . . . . 10 ((a ∩ (ab )) ≡ (ab )) = 1
21 wfh.2 . . . . . . . . . . . 12 C (a, c) = 1
2221wcomcom2 415 . . . . . . . . . . 11 C (a, c ) = 1
2322wcom3ii 419 . . . . . . . . . 10 ((a ∩ (ac )) ≡ (ac )) = 1
2420, 23w2an 373 . . . . . . . . 9 (((a ∩ (ab )) ∩ (a ∩ (ac ))) ≡ ((ab ) ∩ (ac ))) = 1
25 anandi 114 . . . . . . . . . 10 (a ∩ ((ab ) ∩ (ac ))) = ((a ∩ (ab )) ∩ (a ∩ (ac )))
2625bi1 118 . . . . . . . . 9 ((a ∩ ((ab ) ∩ (ac ))) ≡ ((a ∩ (ab )) ∩ (a ∩ (ac )))) = 1
27 anandi 114 . . . . . . . . . 10 (a ∩ (bc )) = ((ab ) ∩ (ac ))
2827bi1 118 . . . . . . . . 9 ((a ∩ (bc )) ≡ ((ab ) ∩ (ac ))) = 1
2924, 26, 28w3tr1 374 . . . . . . . 8 ((a ∩ ((ab ) ∩ (ac ))) ≡ (a ∩ (bc ))) = 1
3029wlan 370 . . . . . . 7 (((bc) ∩ (a ∩ ((ab ) ∩ (ac )))) ≡ ((bc) ∩ (a ∩ (bc )))) = 1
3117, 30wr2 371 . . . . . 6 ((((bc) ∩ a) ∩ ((ab ) ∩ (ac ))) ≡ ((bc) ∩ (a ∩ (bc )))) = 1
32 an12 81 . . . . . . 7 ((bc) ∩ (a ∩ (bc ))) = (a ∩ ((bc) ∩ (bc )))
3332bi1 118 . . . . . 6 (((bc) ∩ (a ∩ (bc ))) ≡ (a ∩ ((bc) ∩ (bc )))) = 1
3431, 33wr2 371 . . . . 5 ((((bc) ∩ a) ∩ ((ab ) ∩ (ac ))) ≡ (a ∩ ((bc) ∩ (bc )))) = 1
3515, 34wr2 371 . . . 4 (((a ∩ (bc)) ∩ ((ab) ∪ (ac)) ) ≡ (a ∩ ((bc) ∩ (bc )))) = 1
36 oran 87 . . . . . . . . . . 11 (bc) = (bc )
3736bi1 118 . . . . . . . . . 10 ((bc) ≡ (bc ) ) = 1
3837wr1 197 . . . . . . . . 9 ((bc ) ≡ (bc)) = 1
3938wcon3 209 . . . . . . . 8 ((bc ) ≡ (bc) ) = 1
4039wlan 370 . . . . . . 7 (((bc) ∩ (bc )) ≡ ((bc) ∩ (bc) )) = 1
41 dff 101 . . . . . . . . 9 0 = ((bc) ∩ (bc) )
4241bi1 118 . . . . . . . 8 (0 ≡ ((bc) ∩ (bc) )) = 1
4342wr1 197 . . . . . . 7 (((bc) ∩ (bc) ) ≡ 0) = 1
4440, 43wr2 371 . . . . . 6 (((bc) ∩ (bc )) ≡ 0) = 1
4544wlan 370 . . . . 5 ((a ∩ ((bc) ∩ (bc ))) ≡ (a ∩ 0)) = 1
46 an0 108 . . . . . 6 (a ∩ 0) = 0
4746bi1 118 . . . . 5 ((a ∩ 0) ≡ 0) = 1
4845, 47wr2 371 . . . 4 ((a ∩ ((bc) ∩ (bc ))) ≡ 0) = 1
4935, 48wr2 371 . . 3 (((a ∩ (bc)) ∩ ((ab) ∪ (ac)) ) ≡ 0) = 1
501, 49wom5 381 . 2 (((ab) ∪ (ac)) ≡ (a ∩ (bc))) = 1
5150wr1 197 1 ((a ∩ (bc)) ≡ ((ab) ∪ (ac))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   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:  wfh3  425  wcom2or  427  wnbdi  429  wlem14  430  ska2  432  wddi1  1105
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