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Theorem u2lemana 606
 Description: Lemma for Dishkant implication study.
Assertion
Ref Expression
u2lemana ((a2 b) ∩ a ) = ((ab) ∪ (ab ))

Proof of Theorem u2lemana
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 . . . . 5 (ab ) C a
6 coman2 186 . . . . . 6 (ab ) C b
76comcom7 460 . . . . 5 (ab ) C b
85, 7fh2r 474 . . . 4 (((ab ) ∪ b) ∩ a ) = (((ab ) ∩ a ) ∪ (ba ))
9 an32 83 . . . . . . 7 ((ab ) ∩ a ) = ((aa ) ∩ b )
10 anidm 111 . . . . . . . 8 (aa ) = a
1110ran 78 . . . . . . 7 ((aa ) ∩ b ) = (ab )
129, 11ax-r2 36 . . . . . 6 ((ab ) ∩ a ) = (ab )
13 ancom 74 . . . . . 6 (ba ) = (ab)
1412, 132or 72 . . . . 5 (((ab ) ∩ a ) ∪ (ba )) = ((ab ) ∪ (ab))
15 ax-a2 31 . . . . 5 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ))
1614, 15ax-r2 36 . . . 4 (((ab ) ∩ a ) ∪ (ba )) = ((ab) ∪ (ab ))
178, 16ax-r2 36 . . 3 (((ab ) ∪ b) ∩ a ) = ((ab) ∪ (ab ))
184, 17ax-r2 36 . 2 ((b ∪ (ab )) ∩ a ) = ((ab) ∪ (ab ))
192, 18ax-r2 36 1 ((a2 b) ∩ a ) = ((ab) ∪ (ab ))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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:  u2lemnoa  661
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