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Theorem wwcomd 214
 Description: Commutation dual (weak). Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wwcomd.1 a C b
Assertion
Ref Expression
wwcomd a = ((ab) ∩ (ab ))

Proof of Theorem wwcomd
StepHypRef Expression
1 wwcomd.1 . . . 4 a C b
21df-c2 133 . . 3 a = ((ab) ∪ (ab ))
3 oran 87 . . . 4 ((ab ) ∪ (ab)) = ((ab ) ∩ (ab) )
4 ax-a2 31 . . . 4 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
5 oran 87 . . . . . 6 (ab) = (ab )
6 anor2 89 . . . . . . . 8 (ab) = (ab )
76ax-r1 35 . . . . . . 7 (ab ) = (ab)
87con3 68 . . . . . 6 (ab ) = (ab)
95, 82an 79 . . . . 5 ((ab) ∩ (ab )) = ((ab ) ∩ (ab) )
109ax-r4 37 . . . 4 ((ab) ∩ (ab )) = ((ab ) ∩ (ab) )
113, 4, 103tr1 63 . . 3 ((ab) ∪ (ab )) = ((ab) ∩ (ab ))
122, 11ax-r2 36 . 2 a = ((ab) ∩ (ab ))
1312con1 66 1 a = ((ab) ∩ (ab ))
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-c2 133 This theorem is referenced by:  wwcom3ii  215
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