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Theorem wnbdi 429
 Description: Negated biconditional (distributive form) (Contributed by NM, 13-Oct-1997.)
Assertion
Ref Expression
wnbdi ((ab) ≡ (((ab) ∩ a ) ∪ ((ab) ∩ b ))) = 1

Proof of Theorem wnbdi
StepHypRef Expression
1 dfnb 95 . . 3 (ab) = ((ab) ∩ (ab ))
21bi1 118 . 2 ((ab) ≡ ((ab) ∩ (ab ))) = 1
3 wcomorr 412 . . . . 5 C (a, (ab)) = 1
43wcomcom 414 . . . 4 C ((ab), a) = 1
54wcomcom2 415 . . 3 C ((ab), a ) = 1
6 wcomorr 412 . . . . . 6 C (b, (ba)) = 1
7 ax-a2 31 . . . . . . 7 (ba) = (ab)
87bi1 118 . . . . . 6 ((ba) ≡ (ab)) = 1
96, 8wcbtr 411 . . . . 5 C (b, (ab)) = 1
109wcomcom 414 . . . 4 C ((ab), b) = 1
1110wcomcom2 415 . . 3 C ((ab), b ) = 1
125, 11wfh1 423 . 2 (((ab) ∩ (ab )) ≡ (((ab) ∩ a ) ∪ ((ab) ∩ b ))) = 1
132, 12wr2 371 1 ((ab) ≡ (((ab) ∩ a ) ∪ ((ab) ∩ b ))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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  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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