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Theorem ud3lem2 571
 Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud3lem2 ((a ∪ (ab )) →3 a) = (ab)

Proof of Theorem ud3lem2
StepHypRef Expression
1 oran 87 . . . . . . 7 (ab) = (ab )
21ax-r1 35 . . . . . 6 (ab ) = (ab)
32con3 68 . . . . 5 (ab ) = (ab)
43lor 70 . . . 4 (a ∪ (ab )) = (a ∪ (ab) )
5 anor2 89 . . . . . 6 (a ∩ (ab)) = (a ∪ (ab) )
65ax-r1 35 . . . . 5 (a ∪ (ab) ) = (a ∩ (ab))
76con3 68 . . . 4 (a ∪ (ab) ) = (a ∩ (ab))
84, 7ax-r2 36 . . 3 (a ∪ (ab )) = (a ∩ (ab))
98ud3lem0b 261 . 2 ((a ∪ (ab )) →3 a) = ((a ∩ (ab))3 a)
10 df-i3 46 . . 3 ((a ∩ (ab))3 a) = ((((a ∩ (ab)) a) ∪ ((a ∩ (ab)) a )) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a)))
11 ax-a3 32 . . . 4 ((((a ∩ (ab)) a) ∪ ((a ∩ (ab)) a )) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) = (((a ∩ (ab)) a) ∪ (((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))))
12 ax-a2 31 . . . . 5 (((a ∩ (ab)) a) ∪ (((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a)))) = ((((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) ∪ ((a ∩ (ab)) a))
13 ax-a1 30 . . . . . . . . . . . . 13 (a ∩ (ab)) = (a ∩ (ab))
1413ran 78 . . . . . . . . . . . 12 ((a ∩ (ab)) ∩ a ) = ((a ∩ (ab)) a )
1514ax-r1 35 . . . . . . . . . . 11 ((a ∩ (ab)) a ) = ((a ∩ (ab)) ∩ a )
16 an32 83 . . . . . . . . . . . 12 ((a ∩ (ab)) ∩ a ) = ((aa ) ∩ (ab))
17 anidm 111 . . . . . . . . . . . . 13 (aa ) = a
1817ran 78 . . . . . . . . . . . 12 ((aa ) ∩ (ab)) = (a ∩ (ab))
1916, 18ax-r2 36 . . . . . . . . . . 11 ((a ∩ (ab)) ∩ a ) = (a ∩ (ab))
2015, 19ax-r2 36 . . . . . . . . . 10 ((a ∩ (ab)) a ) = (a ∩ (ab))
2113ax-r5 38 . . . . . . . . . . . . 13 ((a ∩ (ab)) ∪ a) = ((a ∩ (ab)) a)
227, 212an 79 . . . . . . . . . . . 12 ((a ∪ (ab) ) ∩ ((a ∩ (ab)) ∪ a)) = ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))
2322ax-r1 35 . . . . . . . . . . 11 ((a ∩ (ab)) ∩ ((a ∩ (ab)) a)) = ((a ∪ (ab) ) ∩ ((a ∩ (ab)) ∪ a))
24 ax-a2 31 . . . . . . . . . . . . . 14 ((a ∩ (ab)) ∪ a) = (a ∪ (a ∩ (ab)))
25 oml 445 . . . . . . . . . . . . . 14 (a ∪ (a ∩ (ab))) = (ab)
2624, 25ax-r2 36 . . . . . . . . . . . . 13 ((a ∩ (ab)) ∪ a) = (ab)
2726lan 77 . . . . . . . . . . . 12 ((a ∪ (ab) ) ∩ ((a ∩ (ab)) ∪ a)) = ((a ∪ (ab) ) ∩ (ab))
28 comorr 184 . . . . . . . . . . . . . 14 a C (ab)
2928comcom2 183 . . . . . . . . . . . . . 14 a C (ab)
3028, 29fh2r 474 . . . . . . . . . . . . 13 ((a ∪ (ab) ) ∩ (ab)) = ((a ∩ (ab)) ∪ ((ab) ∩ (ab)))
31 anabs 121 . . . . . . . . . . . . . . 15 (a ∩ (ab)) = a
32 ancom 74 . . . . . . . . . . . . . . . 16 ((ab) ∩ (ab)) = ((ab) ∩ (ab) )
33 dff 101 . . . . . . . . . . . . . . . . 17 0 = ((ab) ∩ (ab) )
3433ax-r1 35 . . . . . . . . . . . . . . . 16 ((ab) ∩ (ab) ) = 0
3532, 34ax-r2 36 . . . . . . . . . . . . . . 15 ((ab) ∩ (ab)) = 0
3631, 352or 72 . . . . . . . . . . . . . 14 ((a ∩ (ab)) ∪ ((ab) ∩ (ab))) = (a ∪ 0)
37 or0 102 . . . . . . . . . . . . . 14 (a ∪ 0) = a
3836, 37ax-r2 36 . . . . . . . . . . . . 13 ((a ∩ (ab)) ∪ ((ab) ∩ (ab))) = a
3930, 38ax-r2 36 . . . . . . . . . . . 12 ((a ∪ (ab) ) ∩ (ab)) = a
4027, 39ax-r2 36 . . . . . . . . . . 11 ((a ∪ (ab) ) ∩ ((a ∩ (ab)) ∪ a)) = a
4123, 40ax-r2 36 . . . . . . . . . 10 ((a ∩ (ab)) ∩ ((a ∩ (ab)) a)) = a
4220, 412or 72 . . . . . . . . 9 (((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) = ((a ∩ (ab)) ∪ a)
4342, 24ax-r2 36 . . . . . . . 8 (((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) = (a ∪ (a ∩ (ab)))
4443, 25ax-r2 36 . . . . . . 7 (((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) = (ab)
45 ancom 74 . . . . . . . 8 ((a ∩ (ab)) a) = (a ∩ (a ∩ (ab)) )
4613lan 77 . . . . . . . . . 10 (a ∩ (a ∩ (ab))) = (a ∩ (a ∩ (ab)) )
4746ax-r1 35 . . . . . . . . 9 (a ∩ (a ∩ (ab)) ) = (a ∩ (a ∩ (ab)))
48 anass 76 . . . . . . . . . . 11 ((aa ) ∩ (ab)) = (a ∩ (a ∩ (ab)))
4948ax-r1 35 . . . . . . . . . 10 (a ∩ (a ∩ (ab))) = ((aa ) ∩ (ab))
50 ancom 74 . . . . . . . . . . 11 ((aa ) ∩ (ab)) = ((ab) ∩ (aa ))
51 dff 101 . . . . . . . . . . . . . 14 0 = (aa )
5251lan 77 . . . . . . . . . . . . 13 ((ab) ∩ 0) = ((ab) ∩ (aa ))
5352ax-r1 35 . . . . . . . . . . . 12 ((ab) ∩ (aa )) = ((ab) ∩ 0)
54 an0 108 . . . . . . . . . . . 12 ((ab) ∩ 0) = 0
5553, 54ax-r2 36 . . . . . . . . . . 11 ((ab) ∩ (aa )) = 0
5650, 55ax-r2 36 . . . . . . . . . 10 ((aa ) ∩ (ab)) = 0
5749, 56ax-r2 36 . . . . . . . . 9 (a ∩ (a ∩ (ab))) = 0
5847, 57ax-r2 36 . . . . . . . 8 (a ∩ (a ∩ (ab)) ) = 0
5945, 58ax-r2 36 . . . . . . 7 ((a ∩ (ab)) a) = 0
6044, 592or 72 . . . . . 6 ((((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) ∪ ((a ∩ (ab)) a)) = ((ab) ∪ 0)
61 or0 102 . . . . . 6 ((ab) ∪ 0) = (ab)
6260, 61ax-r2 36 . . . . 5 ((((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) ∪ ((a ∩ (ab)) a)) = (ab)
6312, 62ax-r2 36 . . . 4 (((a ∩ (ab)) a) ∪ (((a ∩ (ab)) a ) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a)))) = (ab)
6411, 63ax-r2 36 . . 3 ((((a ∩ (ab)) a) ∪ ((a ∩ (ab)) a )) ∪ ((a ∩ (ab)) ∩ ((a ∩ (ab)) a))) = (ab)
6510, 64ax-r2 36 . 2 ((a ∩ (ab))3 a) = (ab)
669, 65ax-r2 36 1 ((a ∪ (ab )) →3 a) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  ud3  597
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