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Theorem u4lemab 613
 Description: Lemma for non-tollens implication study.
Assertion
Ref Expression
u4lemab ((a4 b) ∩ b) = ((ab) ∪ (ab))

Proof of Theorem u4lemab
StepHypRef Expression
1 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
21ran 78 . 2 ((a4 b) ∩ b) = ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∩ b)
3 comanr2 465 . . . . 5 b C (ab)
4 comanr2 465 . . . . 5 b C (ab)
53, 4com2or 483 . . . 4 b C ((ab) ∪ (ab))
6 comanr2 465 . . . . 5 b C ((ab) ∩ b )
76comcom6 459 . . . 4 b C ((ab) ∩ b )
85, 7fh1r 473 . . 3 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∩ b) = ((((ab) ∪ (ab)) ∩ b) ∪ (((ab) ∩ b ) ∩ b))
9 lear 161 . . . . . . 7 (ab) ≤ b
10 lear 161 . . . . . . 7 (ab) ≤ b
119, 10lel2or 170 . . . . . 6 ((ab) ∪ (ab)) ≤ b
1211df2le2 136 . . . . 5 (((ab) ∪ (ab)) ∩ b) = ((ab) ∪ (ab))
13 an32 83 . . . . . 6 (((ab) ∩ b ) ∩ b) = (((ab) ∩ b) ∩ b )
14 anass 76 . . . . . . 7 (((ab) ∩ b) ∩ b ) = ((ab) ∩ (bb ))
15 dff 101 . . . . . . . . . 10 0 = (bb )
1615lan 77 . . . . . . . . 9 ((ab) ∩ 0) = ((ab) ∩ (bb ))
1716ax-r1 35 . . . . . . . 8 ((ab) ∩ (bb )) = ((ab) ∩ 0)
18 an0 108 . . . . . . . 8 ((ab) ∩ 0) = 0
1917, 18ax-r2 36 . . . . . . 7 ((ab) ∩ (bb )) = 0
2014, 19ax-r2 36 . . . . . 6 (((ab) ∩ b) ∩ b ) = 0
2113, 20ax-r2 36 . . . . 5 (((ab) ∩ b ) ∩ b) = 0
2212, 212or 72 . . . 4 ((((ab) ∪ (ab)) ∩ b) ∪ (((ab) ∩ b ) ∩ b)) = (((ab) ∪ (ab)) ∪ 0)
23 or0 102 . . . 4 (((ab) ∪ (ab)) ∪ 0) = ((ab) ∪ (ab))
2422, 23ax-r2 36 . . 3 ((((ab) ∪ (ab)) ∩ b) ∪ (((ab) ∩ b ) ∩ b)) = ((ab) ∪ (ab))
258, 24ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∩ b) = ((ab) ∪ (ab))
262, 25ax-r2 36 1 ((a4 b) ∩ b) = ((ab) ∪ (ab))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u4lemnonb  678  u24lem  770
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