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Theorem u5lem6 769
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u5lem6 (a5 (a5 (a5 b))) = (a5 (a5 b))

Proof of Theorem u5lem6
StepHypRef Expression
1 df-i5 48 . 2 (a5 (a5 (a5 b))) = (((a ∩ (a5 (a5 b))) ∪ (a ∩ (a5 (a5 b)))) ∪ (a ∩ (a5 (a5 b)) ))
2 ancom 74 . . . . 5 ((aa ) ∩ (a5 (a5 b))) = ((a5 (a5 b)) ∩ (aa ))
3 u5lemc1 684 . . . . . . 7 a C (a5 (a5 b))
43comcom 453 . . . . . 6 (a5 (a5 b)) C a
54comcom2 183 . . . . . 6 (a5 (a5 b)) C a
64, 5fh1r 473 . . . . 5 ((aa ) ∩ (a5 (a5 b))) = ((a ∩ (a5 (a5 b))) ∪ (a ∩ (a5 (a5 b))))
7 df-t 41 . . . . . . . 8 1 = (aa )
87ax-r1 35 . . . . . . 7 (aa ) = 1
98lan 77 . . . . . 6 ((a5 (a5 b)) ∩ (aa )) = ((a5 (a5 b)) ∩ 1)
10 an1 106 . . . . . 6 ((a5 (a5 b)) ∩ 1) = (a5 (a5 b))
119, 10ax-r2 36 . . . . 5 ((a5 (a5 b)) ∩ (aa )) = (a5 (a5 b))
122, 6, 113tr2 64 . . . 4 ((a ∩ (a5 (a5 b))) ∪ (a ∩ (a5 (a5 b)))) = (a5 (a5 b))
1312ax-r5 38 . . 3 (((a ∩ (a5 (a5 b))) ∪ (a ∩ (a5 (a5 b)))) ∪ (a ∩ (a5 (a5 b)) )) = ((a5 (a5 b)) ∪ (a ∩ (a5 (a5 b)) ))
143comcom3 454 . . . . 5 a C (a5 (a5 b))
153comcom4 455 . . . . 5 a C (a5 (a5 b))
1614, 15fh4 472 . . . 4 ((a5 (a5 b)) ∪ (a ∩ (a5 (a5 b)) )) = (((a5 (a5 b)) ∪ a ) ∩ ((a5 (a5 b)) ∪ (a5 (a5 b)) ))
17 df-t 41 . . . . . . 7 1 = ((a5 (a5 b)) ∪ (a5 (a5 b)) )
1817ax-r1 35 . . . . . 6 ((a5 (a5 b)) ∪ (a5 (a5 b)) ) = 1
1918lan 77 . . . . 5 (((a5 (a5 b)) ∪ a ) ∩ ((a5 (a5 b)) ∪ (a5 (a5 b)) )) = (((a5 (a5 b)) ∪ a ) ∩ 1)
20 an1 106 . . . . . 6 (((a5 (a5 b)) ∪ a ) ∩ 1) = ((a5 (a5 b)) ∪ a )
21 u5lem5 765 . . . . . . . 8 (a5 (a5 b)) = (a ∪ (ab))
2221ax-r5 38 . . . . . . 7 ((a5 (a5 b)) ∪ a ) = ((a ∪ (ab)) ∪ a )
23 oridm 110 . . . . . . . . 9 (aa ) = a
2423ax-r5 38 . . . . . . . 8 ((aa ) ∪ (ab)) = (a ∪ (ab))
25 or32 82 . . . . . . . 8 ((a ∪ (ab)) ∪ a ) = ((aa ) ∪ (ab))
2624, 25, 213tr1 63 . . . . . . 7 ((a ∪ (ab)) ∪ a ) = (a5 (a5 b))
2722, 26ax-r2 36 . . . . . 6 ((a5 (a5 b)) ∪ a ) = (a5 (a5 b))
2820, 27ax-r2 36 . . . . 5 (((a5 (a5 b)) ∪ a ) ∩ 1) = (a5 (a5 b))
2919, 28ax-r2 36 . . . 4 (((a5 (a5 b)) ∪ a ) ∩ ((a5 (a5 b)) ∪ (a5 (a5 b)) )) = (a5 (a5 b))
3016, 29ax-r2 36 . . 3 ((a5 (a5 b)) ∪ (a ∩ (a5 (a5 b)) )) = (a5 (a5 b))
3113, 30ax-r2 36 . 2 (((a ∩ (a5 (a5 b))) ∪ (a ∩ (a5 (a5 b)))) ∪ (a ∩ (a5 (a5 b)) )) = (a5 (a5 b))
321, 31ax-r2 36 1 (a5 (a5 (a5 b))) = (a5 (a5 b))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →5 wi5 16 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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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