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Theorem ni31 250
 Description: Equivalence for Kalmbach implication.
Assertion
Ref Expression
ni31 (a3 b) = (((ab ) ∩ (ab)) ∩ (a ∪ (ab )))

Proof of Theorem ni31
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
2 oran 87 . . . 4 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (((ab) ∪ (ab )) ∩ (a ∩ (ab)) )
3 oran 87 . . . . . . . 8 ((ab) ∪ (ab )) = ((ab) ∩ (ab ) )
4 anor2 89 . . . . . . . . . . 11 (ab) = (ab )
54con2 67 . . . . . . . . . 10 (ab) = (ab )
6 oran 87 . . . . . . . . . . 11 (ab) = (ab )
76ax-r1 35 . . . . . . . . . 10 (ab ) = (ab)
85, 72an 79 . . . . . . . . 9 ((ab) ∩ (ab ) ) = ((ab ) ∩ (ab))
98ax-r4 37 . . . . . . . 8 ((ab) ∩ (ab ) ) = ((ab ) ∩ (ab))
103, 9ax-r2 36 . . . . . . 7 ((ab) ∪ (ab )) = ((ab ) ∩ (ab))
1110con2 67 . . . . . 6 ((ab) ∪ (ab )) = ((ab ) ∩ (ab))
12 df-a 40 . . . . . . . 8 (a ∩ (ab)) = (a ∪ (ab) )
13 anor1 88 . . . . . . . . . . 11 (ab ) = (ab)
1413ax-r1 35 . . . . . . . . . 10 (ab) = (ab )
1514lor 70 . . . . . . . . 9 (a ∪ (ab) ) = (a ∪ (ab ))
1615ax-r4 37 . . . . . . . 8 (a ∪ (ab) ) = (a ∪ (ab ))
1712, 16ax-r2 36 . . . . . . 7 (a ∩ (ab)) = (a ∪ (ab ))
1817con2 67 . . . . . 6 (a ∩ (ab)) = (a ∪ (ab ))
1911, 182an 79 . . . . 5 (((ab) ∪ (ab )) ∩ (a ∩ (ab)) ) = (((ab ) ∩ (ab)) ∩ (a ∪ (ab )))
2019ax-r4 37 . . . 4 (((ab) ∪ (ab )) ∩ (a ∩ (ab)) ) = (((ab ) ∩ (ab)) ∩ (a ∪ (ab )))
212, 20ax-r2 36 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = (((ab ) ∩ (ab)) ∩ (a ∪ (ab )))
221, 21ax-r2 36 . 2 (a3 b) = (((ab ) ∩ (ab)) ∩ (a ∪ (ab )))
2322con2 67 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:  ud3lem0c  279
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