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Theorem wcomdr 421
 Description: Commutation dual. Kalmbach 83 p. 23.
Hypothesis
Ref Expression
wcomdr.1 (a ≡ ((ab) ∩ (ab ))) = 1
Assertion
Ref Expression
wcomdr C (a, b) = 1

Proof of Theorem wcomdr
StepHypRef Expression
1 wcomdr.1 . . . . 5 (a ≡ ((ab) ∩ (ab ))) = 1
2 df-a 40 . . . . . . 7 ((ab) ∩ (ab )) = ((ab) ∪ (ab ) )
32bi1 118 . . . . . 6 (((ab) ∩ (ab )) ≡ ((ab) ∪ (ab ) ) ) = 1
4 oran 87 . . . . . . . . . 10 (ab) = (ab )
54bi1 118 . . . . . . . . 9 ((ab) ≡ (ab ) ) = 1
65wcon2 208 . . . . . . . 8 ((ab) ≡ (ab )) = 1
7 oran 87 . . . . . . . . . 10 (ab ) = (ab )
87bi1 118 . . . . . . . . 9 ((ab ) ≡ (ab ) ) = 1
98wcon2 208 . . . . . . . 8 ((ab ) ≡ (ab )) = 1
106, 9w2or 372 . . . . . . 7 (((ab) ∪ (ab ) ) ≡ ((ab ) ∪ (ab ))) = 1
1110wr4 199 . . . . . 6 (((ab) ∪ (ab ) ) ≡ ((ab ) ∪ (ab )) ) = 1
123, 11wr2 371 . . . . 5 (((ab) ∩ (ab )) ≡ ((ab ) ∪ (ab )) ) = 1
131, 12wr2 371 . . . 4 (a ≡ ((ab ) ∪ (ab )) ) = 1
1413wcon2 208 . . 3 (a ≡ ((ab ) ∪ (ab ))) = 1
1514wdf-c1 383 . 2 C (a , b ) = 1
1615wcomcom5 420 1 C (a, b) = 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: (None)
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