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Theorem wcomd 418
 Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomcom.1 C (a, b) = 1
Assertion
Ref Expression
wcomd (a ≡ ((ab) ∩ (ab ))) = 1

Proof of Theorem wcomd
StepHypRef Expression
1 wcomcom.1 . . . . 5 C (a, b) = 1
21wcomcom4 417 . . . 4 C (a , b ) = 1
32wdf-c2 384 . . 3 (a ≡ ((ab ) ∪ (ab ))) = 1
43wcon3 209 . 2 (a ≡ ((ab ) ∪ (ab )) ) = 1
5 oran 87 . . . . 5 ((ab ) ∪ (ab )) = ((ab ) ∩ (ab ) )
65bi1 118 . . . 4 (((ab ) ∪ (ab )) ≡ ((ab ) ∩ (ab ) ) ) = 1
76wcon2 208 . . 3 (((ab ) ∪ (ab )) ≡ ((ab ) ∩ (ab ) )) = 1
8 oran 87 . . . . . 6 (ab) = (ab )
98bi1 118 . . . . 5 ((ab) ≡ (ab ) ) = 1
10 oran 87 . . . . . 6 (ab ) = (ab )
1110bi1 118 . . . . 5 ((ab ) ≡ (ab ) ) = 1
129, 11w2an 373 . . . 4 (((ab) ∩ (ab )) ≡ ((ab ) ∩ (ab ) )) = 1
1312wr1 197 . . 3 (((ab ) ∩ (ab ) ) ≡ ((ab) ∩ (ab ))) = 1
147, 13wr2 371 . 2 (((ab ) ∪ (ab )) ≡ ((ab) ∩ (ab ))) = 1
154, 14wr2 371 1 (a ≡ ((ab) ∩ (ab ))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ 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:  wcom3ii  419
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