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Theorem u5lemanb 619
 Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemanb ((a5 b) ∩ b ) = (ab )

Proof of Theorem u5lemanb
StepHypRef Expression
1 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
21ran 78 . 2 ((a5 b) ∩ b ) = ((((ab) ∪ (ab)) ∪ (ab )) ∩ b )
3 comanr2 465 . . . . . 6 b C (ab)
43comcom3 454 . . . . 5 b C (ab)
5 comanr2 465 . . . . . 6 b C (ab)
65comcom3 454 . . . . 5 b C (ab)
74, 6com2or 483 . . . 4 b C ((ab) ∪ (ab))
8 comanr2 465 . . . 4 b C (ab )
97, 8fh1r 473 . . 3 ((((ab) ∪ (ab)) ∪ (ab )) ∩ b ) = ((((ab) ∪ (ab)) ∩ b ) ∪ ((ab ) ∩ b ))
10 ax-a2 31 . . . 4 ((((ab) ∪ (ab)) ∩ b ) ∪ ((ab ) ∩ b )) = (((ab ) ∩ b ) ∪ (((ab) ∪ (ab)) ∩ b ))
11 anass 76 . . . . . . 7 ((ab ) ∩ b ) = (a ∩ (bb ))
12 anidm 111 . . . . . . . 8 (bb ) = b
1312lan 77 . . . . . . 7 (a ∩ (bb )) = (ab )
1411, 13ax-r2 36 . . . . . 6 ((ab ) ∩ b ) = (ab )
154, 6fh1r 473 . . . . . . 7 (((ab) ∪ (ab)) ∩ b ) = (((ab) ∩ b ) ∪ ((ab) ∩ b ))
16 anass 76 . . . . . . . . . 10 ((ab) ∩ b ) = (a ∩ (bb ))
17 dff 101 . . . . . . . . . . . . 13 0 = (bb )
1817lan 77 . . . . . . . . . . . 12 (a ∩ 0) = (a ∩ (bb ))
1918ax-r1 35 . . . . . . . . . . 11 (a ∩ (bb )) = (a ∩ 0)
20 an0 108 . . . . . . . . . . 11 (a ∩ 0) = 0
2119, 20ax-r2 36 . . . . . . . . . 10 (a ∩ (bb )) = 0
2216, 21ax-r2 36 . . . . . . . . 9 ((ab) ∩ b ) = 0
23 anass 76 . . . . . . . . . 10 ((ab) ∩ b ) = (a ∩ (bb ))
2417lan 77 . . . . . . . . . . . 12 (a ∩ 0) = (a ∩ (bb ))
2524ax-r1 35 . . . . . . . . . . 11 (a ∩ (bb )) = (a ∩ 0)
26 an0 108 . . . . . . . . . . 11 (a ∩ 0) = 0
2725, 26ax-r2 36 . . . . . . . . . 10 (a ∩ (bb )) = 0
2823, 27ax-r2 36 . . . . . . . . 9 ((ab) ∩ b ) = 0
2922, 282or 72 . . . . . . . 8 (((ab) ∩ b ) ∪ ((ab) ∩ b )) = (0 ∪ 0)
30 or0 102 . . . . . . . 8 (0 ∪ 0) = 0
3129, 30ax-r2 36 . . . . . . 7 (((ab) ∩ b ) ∪ ((ab) ∩ b )) = 0
3215, 31ax-r2 36 . . . . . 6 (((ab) ∪ (ab)) ∩ b ) = 0
3314, 322or 72 . . . . 5 (((ab ) ∩ b ) ∪ (((ab) ∪ (ab)) ∩ b )) = ((ab ) ∪ 0)
34 or0 102 . . . . 5 ((ab ) ∪ 0) = (ab )
3533, 34ax-r2 36 . . . 4 (((ab ) ∩ b ) ∪ (((ab) ∪ (ab)) ∩ b )) = (ab )
3610, 35ax-r2 36 . . 3 ((((ab) ∪ (ab)) ∩ b ) ∪ ((ab ) ∩ b )) = (ab )
379, 36ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∩ b ) = (ab )
382, 37ax-r2 36 1 ((a5 b) ∩ b ) = (ab )
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →5 wi5 16 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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u5lemnob  674  u5lembi  725
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