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Theorem ud5lem2 590
Description: Lemma for unified disjunction. (Contributed by NM, 10-Apr-2012.)
Assertion
Ref Expression
ud5lem2 ((ab ) →5 a) = (a ∪ (ab))

Proof of Theorem ud5lem2
StepHypRef Expression
1 df-i5 48 . 2 ((ab ) →5 a) = ((((ab ) ∩ a) ∪ ((ab )a)) ∪ ((ab )a ))
2 ax-a3 32 . . 3 ((((ab ) ∩ a) ∪ ((ab )a)) ∪ ((ab )a )) = (((ab ) ∩ a) ∪ (((ab )a) ∪ ((ab )a )))
3 ancom 74 . . . . 5 ((ab ) ∩ a) = (a ∩ (ab ))
4 anabs 121 . . . . 5 (a ∩ (ab )) = a
53, 4ax-r2 36 . . . 4 ((ab ) ∩ a) = a
6 ax-a2 31 . . . . 5 (((ab )a) ∪ ((ab )a )) = (((ab )a ) ∪ ((ab )a))
7 anor2 89 . . . . . . . . . 10 (ab) = (ab )
87ax-r1 35 . . . . . . . . 9 (ab ) = (ab)
98ran 78 . . . . . . . 8 ((ab )a ) = ((ab) ∩ a )
10 an32 83 . . . . . . . . 9 ((ab) ∩ a ) = ((aa ) ∩ b)
11 anidm 111 . . . . . . . . . 10 (aa ) = a
1211ran 78 . . . . . . . . 9 ((aa ) ∩ b) = (ab)
1310, 12ax-r2 36 . . . . . . . 8 ((ab) ∩ a ) = (ab)
149, 13ax-r2 36 . . . . . . 7 ((ab )a ) = (ab)
158ran 78 . . . . . . . 8 ((ab )a) = ((ab) ∩ a)
16 an32 83 . . . . . . . . 9 ((ab) ∩ a) = ((aa) ∩ b)
17 ancom 74 . . . . . . . . . 10 ((aa) ∩ b) = (b ∩ (aa))
18 ancom 74 . . . . . . . . . . . . 13 (aa) = (aa )
19 dff 101 . . . . . . . . . . . . . 14 0 = (aa )
2019ax-r1 35 . . . . . . . . . . . . 13 (aa ) = 0
2118, 20ax-r2 36 . . . . . . . . . . . 12 (aa) = 0
2221lan 77 . . . . . . . . . . 11 (b ∩ (aa)) = (b ∩ 0)
23 an0 108 . . . . . . . . . . 11 (b ∩ 0) = 0
2422, 23ax-r2 36 . . . . . . . . . 10 (b ∩ (aa)) = 0
2517, 24ax-r2 36 . . . . . . . . 9 ((aa) ∩ b) = 0
2616, 25ax-r2 36 . . . . . . . 8 ((ab) ∩ a) = 0
2715, 26ax-r2 36 . . . . . . 7 ((ab )a) = 0
2814, 272or 72 . . . . . 6 (((ab )a ) ∪ ((ab )a)) = ((ab) ∪ 0)
29 or0 102 . . . . . 6 ((ab) ∪ 0) = (ab)
3028, 29ax-r2 36 . . . . 5 (((ab )a ) ∪ ((ab )a)) = (ab)
316, 30ax-r2 36 . . . 4 (((ab )a) ∪ ((ab )a )) = (ab)
325, 312or 72 . . 3 (((ab ) ∩ a) ∪ (((ab )a) ∪ ((ab )a ))) = (a ∪ (ab))
332, 32ax-r2 36 . 2 ((((ab ) ∩ a) ∪ ((ab )a)) ∪ ((ab )a )) = (a ∪ (ab))
341, 33ax-r2 36 1 ((ab ) →5 a) = (a ∪ (ab))
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  0wf 9  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
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i5 48
This theorem is referenced by:  ud5  599
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