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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 ∪ (bc)) ≡ ((ab) ∩ (ac))) = 1

Proof of Theorem wwfh3
StepHypRef Expression
1 conb 122 . . 3 ((a ∪ (bc)) ≡ ((ab) ∩ (ac))) = ((a ∪ (bc)) ≡ ((ab) ∩ (ac)) )
2 oran 87 . . . . . 6 (a ∪ (bc)) = (a ∩ (bc) )
3 df-a 40 . . . . . . . . 9 (bc) = (bc )
43con2 67 . . . . . . . 8 (bc) = (bc )
54lan 77 . . . . . . 7 (a ∩ (bc) ) = (a ∩ (bc ))
65ax-r4 37 . . . . . 6 (a ∩ (bc) ) = (a ∩ (bc ))
72, 6ax-r2 36 . . . . 5 (a ∪ (bc)) = (a ∩ (bc ))
87con2 67 . . . 4 (a ∪ (bc)) = (a ∩ (bc ))
9 df-a 40 . . . . . 6 ((ab) ∩ (ac)) = ((ab) ∪ (ac) )
10 oran 87 . . . . . . . . 9 (ab) = (ab )
1110con2 67 . . . . . . . 8 (ab) = (ab )
12 oran 87 . . . . . . . . 9 (ac) = (ac )
1312con2 67 . . . . . . . 8 (ac) = (ac )
1411, 132or 72 . . . . . . 7 ((ab) ∪ (ac) ) = ((ab ) ∪ (ac ))
1514ax-r4 37 . . . . . 6 ((ab) ∪ (ac) ) = ((ab ) ∪ (ac ))
169, 15ax-r2 36 . . . . 5 ((ab) ∩ (ac)) = ((ab ) ∪ (ac ))
1716con2 67 . . . 4 ((ab) ∩ (ac)) = ((ab ) ∪ (ac ))
188, 172bi 99 . . 3 ((a ∪ (bc)) ≡ ((ab) ∩ (ac)) ) = ((a ∩ (bc )) ≡ ((ab ) ∪ (ac )))
191, 18ax-r2 36 . 2 ((a ∪ (bc)) ≡ ((ab) ∩ (ac))) = ((a ∩ (bc )) ≡ ((ab ) ∪ (ac )))
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 ∩ (bc )) ≡ ((ab ) ∪ (ac ))) = 1
2519, 24ax-r2 36 1 ((a ∪ (bc)) ≡ ((ab) ∩ (ac))) = 1
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ≡ tb 5   ∪ 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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