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Theorem ud2lem2 564
 Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud2lem2 ((a ∪ (ab )) →2 a) = (ab)

Proof of Theorem ud2lem2
StepHypRef Expression
1 df-i2 45 . 2 ((a ∪ (ab )) →2 a) = (a ∪ ((a ∪ (ab ))a ))
2 oran 87 . . . . . . 7 ((a ∪ (ab )) ∪ a) = ((a ∪ (ab ))a )
32con2 67 . . . . . 6 ((a ∪ (ab )) ∪ a) = ((a ∪ (ab ))a )
43ax-r1 35 . . . . 5 ((a ∪ (ab ))a ) = ((a ∪ (ab )) ∪ a)
5 oran 87 . . . . . . . . . . . . 13 (ab) = (ab )
65con2 67 . . . . . . . . . . . 12 (ab) = (ab )
76ax-r1 35 . . . . . . . . . . 11 (ab ) = (ab)
87lor 70 . . . . . . . . . 10 (a ∪ (ab )) = (a ∪ (ab) )
9 anor2 89 . . . . . . . . . . . 12 (a ∩ (ab)) = (a ∪ (ab) )
109ax-r1 35 . . . . . . . . . . 11 (a ∪ (ab) ) = (a ∩ (ab))
1110con3 68 . . . . . . . . . 10 (a ∪ (ab) ) = (a ∩ (ab))
128, 11ax-r2 36 . . . . . . . . 9 (a ∪ (ab )) = (a ∩ (ab))
1312con2 67 . . . . . . . 8 (a ∪ (ab )) = (a ∩ (ab))
1413ran 78 . . . . . . 7 ((a ∪ (ab ))a ) = ((a ∩ (ab)) ∩ a )
15 an32 83 . . . . . . . 8 ((a ∩ (ab)) ∩ a ) = ((aa ) ∩ (ab))
16 anidm 111 . . . . . . . . 9 (aa ) = a
1716ran 78 . . . . . . . 8 ((aa ) ∩ (ab)) = (a ∩ (ab))
1815, 17ax-r2 36 . . . . . . 7 ((a ∩ (ab)) ∩ a ) = (a ∩ (ab))
1914, 18ax-r2 36 . . . . . 6 ((a ∪ (ab ))a ) = (a ∩ (ab))
203, 19ax-r2 36 . . . . 5 ((a ∪ (ab )) ∪ a) = (a ∩ (ab))
214, 20ax-r2 36 . . . 4 ((a ∪ (ab ))a ) = (a ∩ (ab))
2221lor 70 . . 3 (a ∪ ((a ∪ (ab ))a )) = (a ∪ (a ∩ (ab)))
23 oml 445 . . 3 (a ∪ (a ∩ (ab))) = (ab)
2422, 23ax-r2 36 . 2 (a ∪ ((a ∪ (ab ))a )) = (ab)
251, 24ax-r2 36 1 ((a ∪ (ab )) →2 a) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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-i2 45 This theorem is referenced by:  ud2  596
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