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Theorem u1lem3 749
 Description: Lemma for unified implication study. (Contributed by NM, 17-Dec-1997.)
Assertion
Ref Expression
u1lem3 (a1 (b1 a)) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u1lem3
StepHypRef Expression
1 df-i1 44 . 2 (a1 (b1 a)) = (a ∪ (a ∩ (b1 a)))
2 ancom 74 . . . . . . . 8 (ab) = (ba)
3 ancom 74 . . . . . . . 8 (ab ) = (ba)
42, 32or 72 . . . . . . 7 ((ab) ∪ (ab )) = ((ba) ∪ (ba))
5 u1lemab 610 . . . . . . . 8 ((b1 a) ∩ a) = ((ba) ∪ (ba))
65ax-r1 35 . . . . . . 7 ((ba) ∪ (ba)) = ((b1 a) ∩ a)
74, 6ax-r2 36 . . . . . 6 ((ab) ∪ (ab )) = ((b1 a) ∩ a)
8 ancom 74 . . . . . 6 ((b1 a) ∩ a) = (a ∩ (b1 a))
97, 8ax-r2 36 . . . . 5 ((ab) ∪ (ab )) = (a ∩ (b1 a))
109ax-r1 35 . . . 4 (a ∩ (b1 a)) = ((ab) ∪ (ab ))
1110lor 70 . . 3 (a ∪ (a ∩ (b1 a))) = (a ∪ ((ab) ∪ (ab )))
12 id 59 . . 3 (a ∪ ((ab) ∪ (ab ))) = (a ∪ ((ab) ∪ (ab )))
1311, 12ax-r2 36 . 2 (a ∪ (a ∩ (b1 a))) = (a ∪ ((ab) ∪ (ab )))
141, 13ax-r2 36 1 (a1 (b1 a)) = (a ∪ ((ab) ∪ (ab )))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u1lem4  757
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