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Theorem nbdi 486
 Description: Negated biconditional (distributive form)
Assertion
Ref Expression
nbdi (ab) = (((ab) ∩ a ) ∪ ((ab) ∩ b ))

Proof of Theorem nbdi
StepHypRef Expression
1 dfnb 95 . 2 (ab) = ((ab) ∩ (ab ))
2 comorr 184 . . . . 5 a C (ab)
32comcom 453 . . . 4 (ab) C a
43comcom2 183 . . 3 (ab) C a
5 comorr 184 . . . . . 6 b C (ba)
6 ax-a2 31 . . . . . 6 (ba) = (ab)
75, 6cbtr 182 . . . . 5 b C (ab)
87comcom 453 . . . 4 (ab) C b
98comcom2 183 . . 3 (ab) C b
104, 9fh1 469 . 2 ((ab) ∩ (ab )) = (((ab) ∩ a ) ∪ ((ab) ∩ b ))
111, 10ax-r2 36 1 (ab) = (((ab) ∩ a ) ∪ ((ab) ∩ b ))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7 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-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: (None)
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