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Theorem i3abs3 524
 Description: Antecedent absorption. (Contributed by NM, 19-Nov-1997.)
Assertion
Ref Expression
i3abs3 ((a3 b) →3 ((a3 b) →3 a)) = ((a3 b) →3 a)

Proof of Theorem i3abs3
StepHypRef Expression
1 df-t 41 . . . . . . . 8 1 = (aa )
21lan 77 . . . . . . 7 ((a3 b) ∩ 1) = ((a3 b) ∩ (aa ))
3 an1 106 . . . . . . 7 ((a3 b) ∩ 1) = (a3 b)
4 comi31 508 . . . . . . . . . 10 a C (a3 b)
54comcom 453 . . . . . . . . 9 (a3 b) C a
65comcom3 454 . . . . . . . 8 (a3 b) C a
75comcom4 455 . . . . . . . 8 (a3 b) C a
86, 7fh1 469 . . . . . . 7 ((a3 b) ∩ (aa )) = (((a3 b)a) ∪ ((a3 b)a ))
92, 3, 83tr2 64 . . . . . 6 (a3 b) = (((a3 b)a) ∪ ((a3 b)a ))
109ax-r1 35 . . . . 5 (((a3 b)a) ∪ ((a3 b)a )) = (a3 b)
11 comid 187 . . . . . . . 8 (a3 b) C (a3 b)
1211comcom2 183 . . . . . . 7 (a3 b) C (a3 b)
1312, 5fh1 469 . . . . . 6 ((a3 b) ∩ ((a3 b)a)) = (((a3 b) ∩ (a3 b) ) ∪ ((a3 b) ∩ a))
14 ax-a2 31 . . . . . . 7 (0 ∪ ((a3 b) ∩ a)) = (((a3 b) ∩ a) ∪ 0)
15 dff 101 . . . . . . . 8 0 = ((a3 b) ∩ (a3 b) )
1615ax-r5 38 . . . . . . 7 (0 ∪ ((a3 b) ∩ a)) = (((a3 b) ∩ (a3 b) ) ∪ ((a3 b) ∩ a))
17 or0 102 . . . . . . 7 (((a3 b) ∩ a) ∪ 0) = ((a3 b) ∩ a)
1814, 16, 173tr2 64 . . . . . 6 (((a3 b) ∩ (a3 b) ) ∪ ((a3 b) ∩ a)) = ((a3 b) ∩ a)
1913, 18ax-r2 36 . . . . 5 ((a3 b) ∩ ((a3 b)a)) = ((a3 b) ∩ a)
2010, 192or 72 . . . 4 ((((a3 b)a) ∪ ((a3 b)a )) ∪ ((a3 b) ∩ ((a3 b)a))) = ((a3 b) ∪ ((a3 b) ∩ a))
2112, 5fh4 472 . . . . 5 ((a3 b) ∪ ((a3 b) ∩ a)) = (((a3 b) ∪ (a3 b)) ∩ ((a3 b)a))
22 ax-a2 31 . . . . . . . . 9 ((a3 b) ∪ (a3 b)) = ((a3 b) ∪ (a3 b) )
23 df-t 41 . . . . . . . . . 10 1 = ((a3 b) ∪ (a3 b) )
2423ax-r1 35 . . . . . . . . 9 ((a3 b) ∪ (a3 b) ) = 1
2522, 24ax-r2 36 . . . . . . . 8 ((a3 b) ∪ (a3 b)) = 1
2625ran 78 . . . . . . 7 (((a3 b) ∪ (a3 b)) ∩ ((a3 b)a)) = (1 ∩ ((a3 b)a))
27 ancom 74 . . . . . . 7 (1 ∩ ((a3 b)a)) = (((a3 b)a) ∩ 1)
2826, 27ax-r2 36 . . . . . 6 (((a3 b) ∪ (a3 b)) ∩ ((a3 b)a)) = (((a3 b)a) ∩ 1)
29 an1 106 . . . . . 6 (((a3 b)a) ∩ 1) = ((a3 b)a)
3028, 29ax-r2 36 . . . . 5 (((a3 b) ∪ (a3 b)) ∩ ((a3 b)a)) = ((a3 b)a)
3121, 30ax-r2 36 . . . 4 ((a3 b) ∪ ((a3 b) ∩ a)) = ((a3 b)a)
3220, 31ax-r2 36 . . 3 ((((a3 b)a) ∪ ((a3 b)a )) ∪ ((a3 b) ∩ ((a3 b)a))) = ((a3 b)a)
3332ax-r1 35 . 2 ((a3 b)a) = ((((a3 b)a) ∪ ((a3 b)a )) ∪ ((a3 b) ∩ ((a3 b)a)))
34 lem4 511 . 2 ((a3 b) →3 ((a3 b) →3 a)) = ((a3 b)a)
35 df-i3 46 . 2 ((a3 b) →3 a) = ((((a3 b)a) ∪ ((a3 b)a )) ∪ ((a3 b) ∩ ((a3 b)a)))
3633, 34, 353tr1 63 1 ((a3 b) →3 ((a3 b) →3 a)) = ((a3 b) →3 a)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  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:  i3th7  549  i3th8  550
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