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Theorem i3n2 501
 Description: Equivalence for Kalmbach implication.
Assertion
Ref Expression
i3n2 (a3 b ) = ((ab) ∪ ((ab ) ∩ (a ∪ (ab ))))

Proof of Theorem i3n2
StepHypRef Expression
1 df2i3 498 . 2 (a3 b ) = ((a b ) ∪ ((a b ) ∩ (a ∪ (a b ))))
2 ax-a1 30 . . . . 5 a = a
3 ax-a1 30 . . . . 5 b = b
42, 32an 79 . . . 4 (ab) = (a b )
52ax-r5 38 . . . . 5 (ab ) = (a b )
62ran 78 . . . . . 6 (ab ) = (a b )
76lor 70 . . . . 5 (a ∪ (ab )) = (a ∪ (a b ))
85, 72an 79 . . . 4 ((ab ) ∩ (a ∪ (ab ))) = ((a b ) ∩ (a ∪ (a b )))
94, 82or 72 . . 3 ((ab) ∪ ((ab ) ∩ (a ∪ (ab )))) = ((a b ) ∪ ((a b ) ∩ (a ∪ (a b ))))
109ax-r1 35 . 2 ((a b ) ∪ ((a b ) ∩ (a ∪ (a b )))) = ((ab) ∪ ((ab ) ∩ (a ∪ (ab ))))
111, 10ax-r2 36 1 (a3 b ) = ((ab) ∪ ((ab ) ∩ (a ∪ (ab ))))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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