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Theorem u3lem8 783
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem8 (a3 (a3 (a3 b))) = 1

Proof of Theorem u3lem8
StepHypRef Expression
1 comi31 508 . . . 4 a C (a3 (a3 b))
21comcom3 454 . . 3 a C (a3 (a3 b))
32u3lemc4 703 . 2 (a3 (a3 (a3 b))) = (a ∪ (a3 (a3 b)))
4 ax-a1 30 . . . . 5 a = a
54ax-r1 35 . . . 4 a = a
6 u3lem7 774 . . . 4 (a3 (a3 b)) = (a ∪ ((ab) ∪ (ab )))
75, 62or 72 . . 3 (a ∪ (a3 (a3 b))) = (a ∪ (a ∪ ((ab) ∪ (ab ))))
8 ax-a3 32 . . . . 5 ((aa ) ∪ ((ab) ∪ (ab ))) = (a ∪ (a ∪ ((ab) ∪ (ab ))))
98ax-r1 35 . . . 4 (a ∪ (a ∪ ((ab) ∪ (ab )))) = ((aa ) ∪ ((ab) ∪ (ab )))
10 ax-a2 31 . . . . 5 ((aa ) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ (aa ))
11 df-t 41 . . . . . . . 8 1 = (aa )
1211ax-r1 35 . . . . . . 7 (aa ) = 1
1312lor 70 . . . . . 6 (((ab) ∪ (ab )) ∪ (aa )) = (((ab) ∪ (ab )) ∪ 1)
14 or1 104 . . . . . 6 (((ab) ∪ (ab )) ∪ 1) = 1
1513, 14ax-r2 36 . . . . 5 (((ab) ∪ (ab )) ∪ (aa )) = 1
1610, 15ax-r2 36 . . . 4 ((aa ) ∪ ((ab) ∪ (ab ))) = 1
179, 16ax-r2 36 . . 3 (a ∪ (a ∪ ((ab) ∪ (ab )))) = 1
187, 17ax-r2 36 . 2 (a ∪ (a3 (a3 b))) = 1
193, 18ax-r2 36 1 (a3 (a3 (a3 b))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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: (None)
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