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Theorem comd 456
 Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
comcom.1 a C b
Assertion
Ref Expression
comd a = ((ab) ∩ (ab ))

Proof of Theorem comd
StepHypRef Expression
1 comcom.1 . . . . 5 a C b
21comcom4 455 . . . 4 a C b
32df-c2 133 . . 3 a = ((ab ) ∪ (ab ))
43con3 68 . 2 a = ((ab ) ∪ (ab ))
5 oran 87 . . . 4 ((ab ) ∪ (ab )) = ((ab ) ∩ (ab ) )
65con2 67 . . 3 ((ab ) ∪ (ab )) = ((ab ) ∩ (ab ) )
7 oran 87 . . . . 5 (ab) = (ab )
8 oran 87 . . . . 5 (ab ) = (ab )
97, 82an 79 . . . 4 ((ab) ∩ (ab )) = ((ab ) ∩ (ab ) )
109ax-r1 35 . . 3 ((ab ) ∩ (ab ) ) = ((ab) ∩ (ab ))
116, 10ax-r2 36 . 2 ((ab ) ∪ (ab )) = ((ab) ∩ (ab ))
124, 11ax-r2 36 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-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 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:  com3ii  457  gsth  489
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