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Theorem bi3 839
 Description: Chained biconditional. (Contributed by NM, 2-Mar-2000.)
Assertion
Ref Expression
bi3 ((ab) ∩ (bc)) = (((ab) ∩ c) ∪ ((ab ) ∩ c ))

Proof of Theorem bi3
StepHypRef Expression
1 dfb 94 . . 3 (ab) = ((ab) ∪ (ab ))
2 u12lembi 726 . . . 4 ((b1 c) ∩ (c2 b)) = (bc)
32ax-r1 35 . . 3 (bc) = ((b1 c) ∩ (c2 b))
41, 32an 79 . 2 ((ab) ∩ (bc)) = (((ab) ∪ (ab )) ∩ ((b1 c) ∩ (c2 b)))
5 df-i1 44 . . . . . 6 (b1 c) = (b ∪ (bc))
65lan 77 . . . . 5 (((ab) ∪ (ab )) ∩ (b1 c)) = (((ab) ∪ (ab )) ∩ (b ∪ (bc)))
7 lear 161 . . . . . . . 8 (ab ) ≤ b
8 leo 158 . . . . . . . 8 b ≤ (b ∪ (bc))
97, 8letr 137 . . . . . . 7 (ab ) ≤ (b ∪ (bc))
109lecom 180 . . . . . 6 (ab ) C (b ∪ (bc))
11 coman1 185 . . . . . . . 8 (ab ) C a
1211comcom7 460 . . . . . . 7 (ab ) C a
13 coman2 186 . . . . . . . 8 (ab ) C b
1413comcom7 460 . . . . . . 7 (ab ) C b
1512, 14com2an 484 . . . . . 6 (ab ) C (ab)
1610, 15fh2rc 480 . . . . 5 (((ab) ∪ (ab )) ∩ (b ∪ (bc))) = (((ab) ∩ (b ∪ (bc))) ∪ ((ab ) ∩ (b ∪ (bc))))
17 comanr2 465 . . . . . . . . 9 b C (ab)
1817comcom3 454 . . . . . . . 8 b C (ab)
19 comanr1 464 . . . . . . . . 9 b C (bc)
2019comcom3 454 . . . . . . . 8 b C (bc)
2118, 20fh2 470 . . . . . . 7 ((ab) ∩ (b ∪ (bc))) = (((ab) ∩ b ) ∪ ((ab) ∩ (bc)))
22 anass 76 . . . . . . . . 9 ((ab) ∩ b ) = (a ∩ (bb ))
23 dff 101 . . . . . . . . . . 11 0 = (bb )
2423lan 77 . . . . . . . . . 10 (a ∩ 0) = (a ∩ (bb ))
2524ax-r1 35 . . . . . . . . 9 (a ∩ (bb )) = (a ∩ 0)
26 an0 108 . . . . . . . . 9 (a ∩ 0) = 0
2722, 25, 263tr 65 . . . . . . . 8 ((ab) ∩ b ) = 0
28 anass 76 . . . . . . . . . 10 (((ab) ∩ b) ∩ c) = ((ab) ∩ (bc))
2928ax-r1 35 . . . . . . . . 9 ((ab) ∩ (bc)) = (((ab) ∩ b) ∩ c)
30 anass 76 . . . . . . . . . . 11 ((ab) ∩ b) = (a ∩ (bb))
31 anidm 111 . . . . . . . . . . . 12 (bb) = b
3231lan 77 . . . . . . . . . . 11 (a ∩ (bb)) = (ab)
3330, 32ax-r2 36 . . . . . . . . . 10 ((ab) ∩ b) = (ab)
3433ran 78 . . . . . . . . 9 (((ab) ∩ b) ∩ c) = ((ab) ∩ c)
3529, 34ax-r2 36 . . . . . . . 8 ((ab) ∩ (bc)) = ((ab) ∩ c)
3627, 352or 72 . . . . . . 7 (((ab) ∩ b ) ∪ ((ab) ∩ (bc))) = (0 ∪ ((ab) ∩ c))
37 or0r 103 . . . . . . 7 (0 ∪ ((ab) ∩ c)) = ((ab) ∩ c)
3821, 36, 373tr 65 . . . . . 6 ((ab) ∩ (b ∪ (bc))) = ((ab) ∩ c)
3913comcom 453 . . . . . . . 8 b C (ab )
4039, 20fh2 470 . . . . . . 7 ((ab ) ∩ (b ∪ (bc))) = (((ab ) ∩ b ) ∪ ((ab ) ∩ (bc)))
41 anass 76 . . . . . . . . 9 ((ab ) ∩ b ) = (a ∩ (bb ))
42 anidm 111 . . . . . . . . . 10 (bb ) = b
4342lan 77 . . . . . . . . 9 (a ∩ (bb )) = (ab )
4441, 43ax-r2 36 . . . . . . . 8 ((ab ) ∩ b ) = (ab )
45 an4 86 . . . . . . . . 9 ((ab ) ∩ (bc)) = ((ab) ∩ (bc))
46 anass 76 . . . . . . . . 9 ((ab) ∩ (bc)) = (a ∩ (b ∩ (bc)))
4723ran 78 . . . . . . . . . . . . 13 (0 ∩ c) = ((bb ) ∩ c)
4847ax-r1 35 . . . . . . . . . . . 12 ((bb ) ∩ c) = (0 ∩ c)
49 anass 76 . . . . . . . . . . . 12 ((bb ) ∩ c) = (b ∩ (bc))
50 an0r 109 . . . . . . . . . . . 12 (0 ∩ c) = 0
5148, 49, 503tr2 64 . . . . . . . . . . 11 (b ∩ (bc)) = 0
5251lan 77 . . . . . . . . . 10 (a ∩ (b ∩ (bc))) = (a ∩ 0)
53 an0 108 . . . . . . . . . 10 (a ∩ 0) = 0
5452, 53ax-r2 36 . . . . . . . . 9 (a ∩ (b ∩ (bc))) = 0
5545, 46, 543tr 65 . . . . . . . 8 ((ab ) ∩ (bc)) = 0
5644, 552or 72 . . . . . . 7 (((ab ) ∩ b ) ∪ ((ab ) ∩ (bc))) = ((ab ) ∪ 0)
57 or0 102 . . . . . . 7 ((ab ) ∪ 0) = (ab )
5840, 56, 573tr 65 . . . . . 6 ((ab ) ∩ (b ∪ (bc))) = (ab )
5938, 582or 72 . . . . 5 (((ab) ∩ (b ∪ (bc))) ∪ ((ab ) ∩ (b ∪ (bc)))) = (((ab) ∩ c) ∪ (ab ))
606, 16, 593tr 65 . . . 4 (((ab) ∪ (ab )) ∩ (b1 c)) = (((ab) ∩ c) ∪ (ab ))
6160ran 78 . . 3 ((((ab) ∪ (ab )) ∩ (b1 c)) ∩ (c2 b)) = ((((ab) ∩ c) ∪ (ab )) ∩ (c2 b))
62 anass 76 . . 3 ((((ab) ∪ (ab )) ∩ (b1 c)) ∩ (c2 b)) = (((ab) ∪ (ab )) ∩ ((b1 c) ∩ (c2 b)))
63 lear 161 . . . . . . . 8 ((ac) ∩ b) ≤ b
64 leo 158 . . . . . . . 8 b ≤ (b ∪ (cb ))
6563, 64letr 137 . . . . . . 7 ((ac) ∩ b) ≤ (b ∪ (cb ))
66 an32 83 . . . . . . 7 ((ab) ∩ c) = ((ac) ∩ b)
67 df-i2 45 . . . . . . 7 (c2 b) = (b ∪ (cb ))
6865, 66, 67le3tr1 140 . . . . . 6 ((ab) ∩ c) ≤ (c2 b)
6968lecom 180 . . . . 5 ((ab) ∩ c) C (c2 b)
70 anass 76 . . . . . . . . . 10 ((ab) ∩ c) = (a ∩ (bc))
71 lea 160 . . . . . . . . . 10 (a ∩ (bc)) ≤ a
7270, 71bltr 138 . . . . . . . . 9 ((ab) ∩ c) ≤ a
73 leo 158 . . . . . . . . 9 a ≤ (ab)
7472, 73letr 137 . . . . . . . 8 ((ab) ∩ c) ≤ (ab)
75 oran 87 . . . . . . . 8 (ab) = (ab )
7674, 75lbtr 139 . . . . . . 7 ((ab) ∩ c) ≤ (ab )
7776lecom 180 . . . . . 6 ((ab) ∩ c) C (ab )
7877comcom7 460 . . . . 5 ((ab) ∩ c) C (ab )
7969, 78fh2r 474 . . . 4 ((((ab) ∩ c) ∪ (ab )) ∩ (c2 b)) = ((((ab) ∩ c) ∩ (c2 b)) ∪ ((ab ) ∩ (c2 b)))
80 anass 76 . . . . . 6 (((ab) ∩ c) ∩ (c2 b)) = ((ab) ∩ (c ∩ (c2 b)))
81 an4 86 . . . . . 6 ((ab) ∩ (c ∩ (c2 b))) = ((ac) ∩ (b ∩ (c2 b)))
82 ancom 74 . . . . . . . . 9 (b ∩ (c2 b)) = ((c2 b) ∩ b)
83 u2lemab 611 . . . . . . . . 9 ((c2 b) ∩ b) = b
8482, 83ax-r2 36 . . . . . . . 8 (b ∩ (c2 b)) = b
8584lan 77 . . . . . . 7 ((ac) ∩ (b ∩ (c2 b))) = ((ac) ∩ b)
86 an32 83 . . . . . . 7 ((ac) ∩ b) = ((ab) ∩ c)
8785, 86ax-r2 36 . . . . . 6 ((ac) ∩ (b ∩ (c2 b))) = ((ab) ∩ c)
8880, 81, 873tr 65 . . . . 5 (((ab) ∩ c) ∩ (c2 b)) = ((ab) ∩ c)
89 anass 76 . . . . . 6 ((ab ) ∩ (c2 b)) = (a ∩ (b ∩ (c2 b)))
90 ancom 74 . . . . . . . 8 (b ∩ (c2 b)) = ((c2 b) ∩ b )
91 u2lemanb 616 . . . . . . . 8 ((c2 b) ∩ b ) = (cb )
9290, 91ax-r2 36 . . . . . . 7 (b ∩ (c2 b)) = (cb )
9392lan 77 . . . . . 6 (a ∩ (b ∩ (c2 b))) = (a ∩ (cb ))
94 an12 81 . . . . . . 7 (a ∩ (cb )) = (c ∩ (ab ))
95 ancom 74 . . . . . . 7 (c ∩ (ab )) = ((ab ) ∩ c )
9694, 95ax-r2 36 . . . . . 6 (a ∩ (cb )) = ((ab ) ∩ c )
9789, 93, 963tr 65 . . . . 5 ((ab ) ∩ (c2 b)) = ((ab ) ∩ c )
9888, 972or 72 . . . 4 ((((ab) ∩ c) ∩ (c2 b)) ∪ ((ab ) ∩ (c2 b))) = (((ab) ∩ c) ∪ ((ab ) ∩ c ))
9979, 98ax-r2 36 . . 3 ((((ab) ∩ c) ∪ (ab )) ∩ (c2 b)) = (((ab) ∩ c) ∪ ((ab ) ∩ c ))
10061, 62, 993tr2 64 . 2 (((ab) ∪ (ab )) ∩ ((b1 c) ∩ (c2 b))) = (((ab) ∩ c) ∪ ((ab ) ∩ c ))
1014, 100ax-r2 36 1 ((ab) ∩ (bc)) = (((ab) ∩ c) ∪ ((ab ) ∩ c ))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →1 wi1 12   →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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  bi4  840  mlaconj4  844
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