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Theorem u4lemnab 653
 Description: Lemma for non-tollens implication study.
Assertion
Ref Expression
u4lemnab ((a4 b)b) = (((ab ) ∩ (ab )) ∩ b)

Proof of Theorem u4lemnab
StepHypRef Expression
1 u4lemonb 638 . . . 4 ((a4 b) ∪ b ) = (((ab) ∪ (ab)) ∪ b )
2 ax-a2 31 . . . . . 6 ((ab) ∪ (ab)) = ((ab) ∪ (ab))
3 anor2 89 . . . . . . . 8 (ab) = (ab )
4 df-a 40 . . . . . . . 8 (ab) = (ab )
53, 42or 72 . . . . . . 7 ((ab) ∪ (ab)) = ((ab ) ∪ (ab ) )
6 oran3 93 . . . . . . 7 ((ab ) ∪ (ab ) ) = ((ab ) ∩ (ab ))
75, 6ax-r2 36 . . . . . 6 ((ab) ∪ (ab)) = ((ab ) ∩ (ab ))
82, 7ax-r2 36 . . . . 5 ((ab) ∪ (ab)) = ((ab ) ∩ (ab ))
98ax-r5 38 . . . 4 (((ab) ∪ (ab)) ∪ b ) = (((ab ) ∩ (ab ))b )
101, 9ax-r2 36 . . 3 ((a4 b) ∪ b ) = (((ab ) ∩ (ab ))b )
11 oran1 91 . . 3 ((a4 b) ∪ b ) = ((a4 b)b)
12 oran3 93 . . 3 (((ab ) ∩ (ab ))b ) = (((ab ) ∩ (ab )) ∩ b)
1310, 11, 123tr2 64 . 2 ((a4 b)b) = (((ab ) ∩ (ab )) ∩ b)
1413con1 66 1 ((a4 b)b) = (((ab ) ∩ (ab )) ∩ b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →4 wi4 15 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i4 47  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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