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Theorem u3lem3 751
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem3 (a3 (b3 a)) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u3lem3
StepHypRef Expression
1 df-i3 46 . 2 (a3 (b3 a)) = (((a ∩ (b3 a)) ∪ (a ∩ (b3 a) )) ∪ (a ∩ (a ∪ (b3 a))))
2 ancom 74 . . . . . . 7 (a ∩ (b3 a)) = ((b3 a) ∩ a )
3 u3lemanb 617 . . . . . . 7 ((b3 a) ∩ a ) = (ba )
42, 3ax-r2 36 . . . . . 6 (a ∩ (b3 a)) = (ba )
5 ancom 74 . . . . . . 7 (a ∩ (b3 a) ) = ((b3 a)a )
6 u3lemnanb 657 . . . . . . 7 ((b3 a)a ) = (ba )
75, 6ax-r2 36 . . . . . 6 (a ∩ (b3 a) ) = (ba )
84, 72or 72 . . . . 5 ((a ∩ (b3 a)) ∪ (a ∩ (b3 a) )) = ((ba ) ∪ (ba ))
9 ancom 74 . . . . . . 7 (ba ) = (ab )
10 ancom 74 . . . . . . 7 (ba ) = (ab)
119, 102or 72 . . . . . 6 ((ba ) ∪ (ba )) = ((ab ) ∪ (ab))
12 ax-a2 31 . . . . . 6 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
1311, 12ax-r2 36 . . . . 5 ((ba ) ∪ (ba )) = ((ab) ∪ (ab ))
148, 13ax-r2 36 . . . 4 ((a ∩ (b3 a)) ∪ (a ∩ (b3 a) )) = ((ab) ∪ (ab ))
15 ax-a2 31 . . . . . . 7 (a ∪ (b3 a)) = ((b3 a) ∪ a )
16 u3lemonb 637 . . . . . . 7 ((b3 a) ∪ a ) = 1
1715, 16ax-r2 36 . . . . . 6 (a ∪ (b3 a)) = 1
1817lan 77 . . . . 5 (a ∩ (a ∪ (b3 a))) = (a ∩ 1)
19 an1 106 . . . . 5 (a ∩ 1) = a
2018, 19ax-r2 36 . . . 4 (a ∩ (a ∪ (b3 a))) = a
2114, 202or 72 . . 3 (((a ∩ (b3 a)) ∪ (a ∩ (b3 a) )) ∪ (a ∩ (a ∪ (b3 a)))) = (((ab) ∪ (ab )) ∪ a)
22 ax-a2 31 . . 3 (((ab) ∪ (ab )) ∪ a) = (a ∪ ((ab) ∪ (ab )))
2321, 22ax-r2 36 . 2 (((a ∩ (b3 a)) ∪ (a ∩ (b3 a) )) ∪ (a ∩ (a ∪ (b3 a)))) = (a ∪ ((ab) ∪ (ab )))
241, 23ax-r2 36 1 (a3 (b3 a)) = (a ∪ ((ab) ∪ (ab )))
 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:  u3lem3n  754  u3lem14a  791
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