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Theorem i3abs1 522
 Description: Antecedent absorption. (Contributed by NM, 16-Nov-1997.)
Assertion
Ref Expression
i3abs1 (a3 (a3 (a3 b))) = (a3 (a3 b))

Proof of Theorem i3abs1
StepHypRef Expression
1 orordi 112 . . . . 5 (a ∪ ((ab) ∪ (ab ))) = ((a ∪ (ab)) ∪ (a ∪ (ab )))
2 orabs 120 . . . . . . 7 (a ∪ (ab)) = a
3 orabs 120 . . . . . . 7 (a ∪ (ab )) = a
42, 32or 72 . . . . . 6 ((a ∪ (ab)) ∪ (a ∪ (ab ))) = (aa )
5 oridm 110 . . . . . 6 (aa ) = a
64, 5ax-r2 36 . . . . 5 ((a ∪ (ab)) ∪ (a ∪ (ab ))) = a
71, 6ax-r2 36 . . . 4 (a ∪ ((ab) ∪ (ab ))) = a
87ax-r5 38 . . 3 ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∩ (ab))) = (a ∪ (a ∩ (ab)))
9 ax-a3 32 . . 3 ((a ∪ ((ab) ∪ (ab ))) ∪ (a ∩ (ab))) = (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
10 omln 446 . . 3 (a ∪ (a ∩ (ab))) = (ab)
118, 9, 103tr2 64 . 2 (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab)))) = (ab)
12 lem4 511 . . 3 (a3 (a3 (a3 b))) = (a ∪ (a3 b))
13 df-i3 46 . . . 4 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
1413lor 70 . . 3 (a ∪ (a3 b)) = (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
1512, 14ax-r2 36 . 2 (a3 (a3 (a3 b))) = (a ∪ (((ab) ∪ (ab )) ∪ (a ∩ (ab))))
16 lem4 511 . 2 (a3 (a3 b)) = (ab)
1711, 15, 163tr1 63 1 (a3 (a3 (a3 b))) = (a3 (a3 b))
 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:  i3abs2  523  i3th6  548
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