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Theorem ud3lem3c 574
 Description: Lemma for unified disjunction. (Contributed by NM, 27-Nov-1997.)
Assertion
Ref Expression
ud3lem3c ((a3 b) ∪ (ab)) = (ab)

Proof of Theorem ud3lem3c
StepHypRef Expression
1 ud3lem0c 279 . . . 4 (a3 b) = (((ab ) ∩ (ab)) ∩ (a ∪ (ab )))
2 an32 83 . . . . 5 (((ab ) ∩ (ab)) ∩ (a ∪ (ab ))) = (((ab ) ∩ (a ∪ (ab ))) ∩ (ab))
3 ancom 74 . . . . 5 (((ab ) ∩ (a ∪ (ab ))) ∩ (ab)) = ((ab) ∩ ((ab ) ∩ (a ∪ (ab ))))
42, 3ax-r2 36 . . . 4 (((ab ) ∩ (ab)) ∩ (a ∪ (ab ))) = ((ab) ∩ ((ab ) ∩ (a ∪ (ab ))))
51, 4ax-r2 36 . . 3 (a3 b) = ((ab) ∩ ((ab ) ∩ (a ∪ (ab ))))
65ax-r5 38 . 2 ((a3 b) ∪ (ab)) = (((ab) ∩ ((ab ) ∩ (a ∪ (ab )))) ∪ (ab))
7 ax-a2 31 . . 3 (((ab) ∩ ((ab ) ∩ (a ∪ (ab )))) ∪ (ab)) = ((ab) ∪ ((ab) ∩ ((ab ) ∩ (a ∪ (ab )))))
8 orabs 120 . . 3 ((ab) ∪ ((ab) ∩ ((ab ) ∩ (a ∪ (ab ))))) = (ab)
97, 8ax-r2 36 . 2 (((ab) ∩ ((ab ) ∩ (a ∪ (ab )))) ∪ (ab)) = (ab)
106, 9ax-r2 36 1 ((a3 b) ∪ (ab)) = (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-a5 34  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:  ud3lem3d  575
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