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Theorem u2lemaa 601
 Description: Lemma for Dishkant implication study.
Assertion
Ref Expression
u2lemaa ((a2 b) ∩ a) = (ab)

Proof of Theorem u2lemaa
StepHypRef Expression
1 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
21ran 78 . 2 ((a2 b) ∩ a) = ((b ∪ (ab )) ∩ a)
3 ax-a2 31 . . . 4 (b ∪ (ab )) = ((ab ) ∪ b)
43ran 78 . . 3 ((b ∪ (ab )) ∩ a) = (((ab ) ∪ b) ∩ a)
5 coman1 185 . . . . . 6 (ab ) C a
65comcom7 460 . . . . 5 (ab ) C a
7 coman2 186 . . . . . 6 (ab ) C b
87comcom7 460 . . . . 5 (ab ) C b
96, 8fh2r 474 . . . 4 (((ab ) ∪ b) ∩ a) = (((ab ) ∩ a) ∪ (ba))
10 ax-a2 31 . . . . 5 (((ab ) ∩ a) ∪ (ba)) = ((ba) ∪ ((ab ) ∩ a))
11 ancom 74 . . . . . . 7 (ba) = (ab)
12 ancom 74 . . . . . . . 8 ((ab ) ∩ a) = (a ∩ (ab ))
13 anass 76 . . . . . . . . . 10 ((aa ) ∩ b ) = (a ∩ (ab ))
1413ax-r1 35 . . . . . . . . 9 (a ∩ (ab )) = ((aa ) ∩ b )
15 ancom 74 . . . . . . . . . 10 ((aa ) ∩ b ) = (b ∩ (aa ))
16 dff 101 . . . . . . . . . . . . 13 0 = (aa )
1716ax-r1 35 . . . . . . . . . . . 12 (aa ) = 0
1817lan 77 . . . . . . . . . . 11 (b ∩ (aa )) = (b ∩ 0)
19 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
2018, 19ax-r2 36 . . . . . . . . . 10 (b ∩ (aa )) = 0
2115, 20ax-r2 36 . . . . . . . . 9 ((aa ) ∩ b ) = 0
2214, 21ax-r2 36 . . . . . . . 8 (a ∩ (ab )) = 0
2312, 22ax-r2 36 . . . . . . 7 ((ab ) ∩ a) = 0
2411, 232or 72 . . . . . 6 ((ba) ∪ ((ab ) ∩ a)) = ((ab) ∪ 0)
25 or0 102 . . . . . 6 ((ab) ∪ 0) = (ab)
2624, 25ax-r2 36 . . . . 5 ((ba) ∪ ((ab ) ∩ a)) = (ab)
2710, 26ax-r2 36 . . . 4 (((ab ) ∩ a) ∪ (ba)) = (ab)
289, 27ax-r2 36 . . 3 (((ab ) ∪ b) ∩ a) = (ab)
294, 28ax-r2 36 . 2 ((b ∪ (ab )) ∩ a) = (ab)
302, 29ax-r2 36 1 ((a2 b) ∩ a) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u2lemnona  666
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