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Theorem u4lemaa 603
Description: Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)
Assertion
Ref Expression
u4lemaa ((a4 b) ∩ a) = (ab)

Proof of Theorem u4lemaa
StepHypRef Expression
1 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
21ran 78 . 2 ((a4 b) ∩ a) = ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∩ a)
3 comanr1 464 . . . . . 6 a C (ab)
4 comanr1 464 . . . . . . 7 a C (ab)
54comcom6 459 . . . . . 6 a C (ab)
63, 5com2or 483 . . . . 5 a C ((ab) ∪ (ab))
76comcom 453 . . . 4 ((ab) ∪ (ab)) C a
83comcom3 454 . . . . . . . 8 a C (ab)
98, 4com2or 483 . . . . . . 7 a C ((ab) ∪ (ab))
109comcom 453 . . . . . 6 ((ab) ∪ (ab)) C a
11 comanr2 465 . . . . . . . 8 b C (ab)
12 comanr2 465 . . . . . . . 8 b C (ab)
1311, 12com2or 483 . . . . . . 7 b C ((ab) ∪ (ab))
1413comcom 453 . . . . . 6 ((ab) ∪ (ab)) C b
1510, 14com2or 483 . . . . 5 ((ab) ∪ (ab)) C (ab)
1614comcom2 183 . . . . 5 ((ab) ∪ (ab)) C b
1715, 16com2an 484 . . . 4 ((ab) ∪ (ab)) C ((ab) ∩ b )
187, 17fh2r 474 . . 3 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∩ a) = ((((ab) ∪ (ab)) ∩ a) ∪ (((ab) ∩ b ) ∩ a))
193, 5fh1r 473 . . . . . 6 (((ab) ∪ (ab)) ∩ a) = (((ab) ∩ a) ∪ ((ab) ∩ a))
20 an32 83 . . . . . . . . 9 ((ab) ∩ a) = ((aa) ∩ b)
21 anidm 111 . . . . . . . . . 10 (aa) = a
2221ran 78 . . . . . . . . 9 ((aa) ∩ b) = (ab)
2320, 22ax-r2 36 . . . . . . . 8 ((ab) ∩ a) = (ab)
24 ancom 74 . . . . . . . . 9 ((ab) ∩ a) = (a ∩ (ab))
25 anass 76 . . . . . . . . . . 11 ((aa ) ∩ b) = (a ∩ (ab))
2625ax-r1 35 . . . . . . . . . 10 (a ∩ (ab)) = ((aa ) ∩ b)
27 ancom 74 . . . . . . . . . . 11 ((aa ) ∩ b) = (b ∩ (aa ))
28 dff 101 . . . . . . . . . . . . . 14 0 = (aa )
2928ax-r1 35 . . . . . . . . . . . . 13 (aa ) = 0
3029lan 77 . . . . . . . . . . . 12 (b ∩ (aa )) = (b ∩ 0)
31 an0 108 . . . . . . . . . . . 12 (b ∩ 0) = 0
3230, 31ax-r2 36 . . . . . . . . . . 11 (b ∩ (aa )) = 0
3327, 32ax-r2 36 . . . . . . . . . 10 ((aa ) ∩ b) = 0
3426, 33ax-r2 36 . . . . . . . . 9 (a ∩ (ab)) = 0
3524, 34ax-r2 36 . . . . . . . 8 ((ab) ∩ a) = 0
3623, 352or 72 . . . . . . 7 (((ab) ∩ a) ∪ ((ab) ∩ a)) = ((ab) ∪ 0)
37 or0 102 . . . . . . 7 ((ab) ∪ 0) = (ab)
3836, 37ax-r2 36 . . . . . 6 (((ab) ∩ a) ∪ ((ab) ∩ a)) = (ab)
3919, 38ax-r2 36 . . . . 5 (((ab) ∪ (ab)) ∩ a) = (ab)
40 anass 76 . . . . . 6 (((ab) ∩ b ) ∩ a) = ((ab) ∩ (ba))
41 ancom 74 . . . . . . . . 9 (ba) = (ab )
42 anor1 88 . . . . . . . . 9 (ab ) = (ab)
4341, 42ax-r2 36 . . . . . . . 8 (ba) = (ab)
4443lan 77 . . . . . . 7 ((ab) ∩ (ba)) = ((ab) ∩ (ab) )
45 dff 101 . . . . . . . 8 0 = ((ab) ∩ (ab) )
4645ax-r1 35 . . . . . . 7 ((ab) ∩ (ab) ) = 0
4744, 46ax-r2 36 . . . . . 6 ((ab) ∩ (ba)) = 0
4840, 47ax-r2 36 . . . . 5 (((ab) ∩ b ) ∩ a) = 0
4939, 482or 72 . . . 4 ((((ab) ∪ (ab)) ∩ a) ∪ (((ab) ∩ b ) ∩ a)) = ((ab) ∪ 0)
5049, 37ax-r2 36 . . 3 ((((ab) ∪ (ab)) ∩ a) ∪ (((ab) ∩ b ) ∩ a)) = (ab)
5118, 50ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∩ a) = (ab)
522, 51ax-r2 36 1 ((a4 b) ∩ a) = (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:  u4lemnona  668  u4lem1  737  u4lem5  764
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