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Theorem u3lem7 774
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem7 (a3 (a3 b)) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u3lem7
StepHypRef Expression
1 comi31 508 . . . 4 a C (a3 b)
21comcom6 459 . . 3 a C (a3 b)
32u3lemc4 703 . 2 (a3 (a3 b)) = (a ∪ (a3 b))
4 df-i3 46 . . . 4 (a3 b) = (((a b) ∪ (a b )) ∪ (a ∩ (a b)))
54lor 70 . . 3 (a ∪ (a3 b)) = (a ∪ (((a b) ∪ (a b )) ∪ (a ∩ (a b))))
6 or12 80 . . . 4 (a ∪ (((a b) ∪ (a b )) ∪ (a ∩ (a b)))) = (((a b) ∪ (a b )) ∪ (a ∪ (a ∩ (a b))))
7 ax-a1 30 . . . . . . . . 9 a = a
87ran 78 . . . . . . . 8 (ab) = (a b)
97ran 78 . . . . . . . 8 (ab ) = (a b )
108, 92or 72 . . . . . . 7 ((ab) ∪ (ab )) = ((a b) ∪ (a b ))
1110ax-r1 35 . . . . . 6 ((a b) ∪ (a b )) = ((ab) ∪ (ab ))
12 orabs 120 . . . . . 6 (a ∪ (a ∩ (a b))) = a
1311, 122or 72 . . . . 5 (((a b) ∪ (a b )) ∪ (a ∪ (a ∩ (a b)))) = (((ab) ∪ (ab )) ∪ a )
14 ax-a2 31 . . . . 5 (((ab) ∪ (ab )) ∪ a ) = (a ∪ ((ab) ∪ (ab )))
1513, 14ax-r2 36 . . . 4 (((a b) ∪ (a b )) ∪ (a ∪ (a ∩ (a b)))) = (a ∪ ((ab) ∪ (ab )))
166, 15ax-r2 36 . . 3 (a ∪ (((a b) ∪ (a b )) ∪ (a ∩ (a b)))) = (a ∪ ((ab) ∪ (ab )))
175, 16ax-r2 36 . 2 (a ∪ (a3 b)) = (a ∪ ((ab) ∪ (ab )))
183, 17ax-r2 36 1 (a3 (a3 b)) = (a ∪ ((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-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:  u3lem8  783  u3lem9  784
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