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Theorem i3n1 249
 Description: Equivalence for Kalmbach implication. (Contributed by NM, 9-Nov-1997.)
Assertion
Ref Expression
i3n1 (a3 b ) = (((ab ) ∪ (ab)) ∪ (a ∩ (ab )))

Proof of Theorem i3n1
StepHypRef Expression
1 df-i3 46 . 2 (a3 b ) = (((a b ) ∪ (a b )) ∪ (a ∩ (a b )))
2 ax-a1 30 . . . . . 6 a = a
32ran 78 . . . . 5 (ab ) = (a b )
4 ax-a1 30 . . . . . 6 b = b
52, 42an 79 . . . . 5 (ab) = (a b )
63, 52or 72 . . . 4 ((ab ) ∪ (ab)) = ((a b ) ∪ (a b ))
72ax-r5 38 . . . . 5 (ab ) = (a b )
87lan 77 . . . 4 (a ∩ (ab )) = (a ∩ (a b ))
96, 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-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i3 46 This theorem is referenced by:  oi3ai3  503  i3con  551  i3orlem7  558  i3orlem8  559
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