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Theorem u1lemanb 615
 Description: Lemma for Sasaki implication study.
Assertion
Ref Expression
u1lemanb ((a1 b) ∩ b ) = (ab )

Proof of Theorem u1lemanb
StepHypRef Expression
1 df-i1 44 . . 3 (a1 b) = (a ∪ (ab))
21ran 78 . 2 ((a1 b) ∩ b ) = ((a ∪ (ab)) ∩ b )
3 ax-a2 31 . . . 4 (a ∪ (ab)) = ((ab) ∪ a )
43ran 78 . . 3 ((a ∪ (ab)) ∩ b ) = (((ab) ∪ a ) ∩ b )
5 coman2 186 . . . . . 6 (ab) C b
65comcom2 183 . . . . 5 (ab) C b
7 coman1 185 . . . . . 6 (ab) C a
87comcom2 183 . . . . 5 (ab) C a
96, 8fh2r 474 . . . 4 (((ab) ∪ a ) ∩ b ) = (((ab) ∩ b ) ∪ (ab ))
10 ax-a2 31 . . . . 5 (((ab) ∩ b ) ∪ (ab )) = ((ab ) ∪ ((ab) ∩ b ))
11 anass 76 . . . . . . . 8 ((ab) ∩ b ) = (a ∩ (bb ))
12 dff 101 . . . . . . . . . . 11 0 = (bb )
1312lan 77 . . . . . . . . . 10 (a ∩ 0) = (a ∩ (bb ))
1413ax-r1 35 . . . . . . . . 9 (a ∩ (bb )) = (a ∩ 0)
15 an0 108 . . . . . . . . 9 (a ∩ 0) = 0
1614, 15ax-r2 36 . . . . . . . 8 (a ∩ (bb )) = 0
1711, 16ax-r2 36 . . . . . . 7 ((ab) ∩ b ) = 0
1817lor 70 . . . . . 6 ((ab ) ∪ ((ab) ∩ b )) = ((ab ) ∪ 0)
19 or0 102 . . . . . 6 ((ab ) ∪ 0) = (ab )
2018, 19ax-r2 36 . . . . 5 ((ab ) ∪ ((ab) ∩ b )) = (ab )
2110, 20ax-r2 36 . . . 4 (((ab) ∩ b ) ∪ (ab )) = (ab )
229, 21ax-r2 36 . . 3 (((ab) ∪ a ) ∩ b ) = (ab )
234, 22ax-r2 36 . 2 ((a ∪ (ab)) ∩ b ) = (ab )
242, 23ax-r2 36 1 ((a1 b) ∩ b ) = (ab )
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u1lemnob  670  u3lem14a  791  negantlem5  853
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