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Theorem ni32 502
 Description: Equivalence for Kalmbach implication. (Contributed by NM, 9-Nov-1997.)
Assertion
Ref Expression
ni32 (a3 b) = ((ab) ∩ ((ab ) ∪ (a ∩ (ab ))))

Proof of Theorem ni32
StepHypRef Expression
1 df2i3 498 . . 3 (a3 b) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))
2 oran 87 . . . 4 ((ab ) ∪ ((ab) ∩ (a ∪ (ab)))) = ((ab ) ∩ ((ab) ∩ (a ∪ (ab))) )
3 oran 87 . . . . . . 7 (ab) = (ab )
4 oran 87 . . . . . . . 8 ((ab ) ∪ (a ∩ (ab ))) = ((ab ) ∩ (a ∩ (ab )) )
5 anor1 88 . . . . . . . . . . . . 13 (ab ) = (ab)
65con2 67 . . . . . . . . . . . 12 (ab ) = (ab)
76ax-r1 35 . . . . . . . . . . 11 (ab) = (ab )
8 oran 87 . . . . . . . . . . . 12 (a ∪ (ab)) = (a ∩ (ab) )
9 anor2 89 . . . . . . . . . . . . . . 15 (ab) = (ab )
109con2 67 . . . . . . . . . . . . . 14 (ab) = (ab )
1110lan 77 . . . . . . . . . . . . 13 (a ∩ (ab) ) = (a ∩ (ab ))
1211ax-r4 37 . . . . . . . . . . . 12 (a ∩ (ab) ) = (a ∩ (ab ))
138, 12ax-r2 36 . . . . . . . . . . 11 (a ∪ (ab)) = (a ∩ (ab ))
147, 132an 79 . . . . . . . . . 10 ((ab) ∩ (a ∪ (ab))) = ((ab ) ∩ (a ∩ (ab )) )
1514ax-r1 35 . . . . . . . . 9 ((ab ) ∩ (a ∩ (ab )) ) = ((ab) ∩ (a ∪ (ab)))
1615ax-r4 37 . . . . . . . 8 ((ab ) ∩ (a ∩ (ab )) ) = ((ab) ∩ (a ∪ (ab)))
174, 16ax-r2 36 . . . . . . 7 ((ab ) ∪ (a ∩ (ab ))) = ((ab) ∩ (a ∪ (ab)))
183, 172an 79 . . . . . 6 ((ab) ∩ ((ab ) ∪ (a ∩ (ab )))) = ((ab ) ∩ ((ab) ∩ (a ∪ (ab))) )
1918ax-r1 35 . . . . 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-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:  oi3ai3  503  i3con  551
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