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Theorem u5lem5 765
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u5lem5 (a5 (a5 b)) = (a ∪ (ab))

Proof of Theorem u5lem5
StepHypRef Expression
1 df-i5 48 . 2 (a5 (a5 b)) = (((a ∩ (a5 b)) ∪ (a ∩ (a5 b))) ∪ (a ∩ (a5 b) ))
2 u5lemc1 684 . . . . . . . 8 a C (a5 b)
32comcom 453 . . . . . . 7 (a5 b) C a
43comcom2 183 . . . . . . 7 (a5 b) C a
53, 4fh1r 473 . . . . . 6 ((aa ) ∩ (a5 b)) = ((a ∩ (a5 b)) ∪ (a ∩ (a5 b)))
65ax-r1 35 . . . . 5 ((a ∩ (a5 b)) ∪ (a ∩ (a5 b))) = ((aa ) ∩ (a5 b))
7 ancom 74 . . . . . 6 ((aa ) ∩ (a5 b)) = ((a5 b) ∩ (aa ))
8 df-t 41 . . . . . . . . 9 1 = (aa )
98ax-r1 35 . . . . . . . 8 (aa ) = 1
109lan 77 . . . . . . 7 ((a5 b) ∩ (aa )) = ((a5 b) ∩ 1)
11 an1 106 . . . . . . 7 ((a5 b) ∩ 1) = (a5 b)
1210, 11ax-r2 36 . . . . . 6 ((a5 b) ∩ (aa )) = (a5 b)
137, 12ax-r2 36 . . . . 5 ((aa ) ∩ (a5 b)) = (a5 b)
146, 13ax-r2 36 . . . 4 ((a ∩ (a5 b)) ∪ (a ∩ (a5 b))) = (a5 b)
1514ax-r5 38 . . 3 (((a ∩ (a5 b)) ∪ (a ∩ (a5 b))) ∪ (a ∩ (a5 b) )) = ((a5 b) ∪ (a ∩ (a5 b) ))
162comcom3 454 . . . . 5 a C (a5 b)
172comcom4 455 . . . . 5 a C (a5 b)
1816, 17fh4 472 . . . 4 ((a5 b) ∪ (a ∩ (a5 b) )) = (((a5 b) ∪ a ) ∩ ((a5 b) ∪ (a5 b) ))
19 df-t 41 . . . . . . 7 1 = ((a5 b) ∪ (a5 b) )
2019ax-r1 35 . . . . . 6 ((a5 b) ∪ (a5 b) ) = 1
2120lan 77 . . . . 5 (((a5 b) ∪ a ) ∩ ((a5 b) ∪ (a5 b) )) = (((a5 b) ∪ a ) ∩ 1)
22 an1 106 . . . . . 6 (((a5 b) ∪ a ) ∩ 1) = ((a5 b) ∪ a )
23 u5lemona 629 . . . . . 6 ((a5 b) ∪ a ) = (a ∪ (ab))
2422, 23ax-r2 36 . . . . 5 (((a5 b) ∪ a ) ∩ 1) = (a ∪ (ab))
2521, 24ax-r2 36 . . . 4 (((a5 b) ∪ a ) ∩ ((a5 b) ∪ (a5 b) )) = (a ∪ (ab))
2618, 25ax-r2 36 . . 3 ((a5 b) ∪ (a ∩ (a5 b) )) = (a ∪ (ab))
2715, 26ax-r2 36 . 2 (((a ∩ (a5 b)) ∪ (a ∩ (a5 b))) ∪ (a ∩ (a5 b) )) = (a ∪ (ab))
281, 27ax-r2 36 1 (a5 (a5 b)) = (a ∪ (ab))
 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:  u5lem6  769
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