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Theorem ud2lem2 564
Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud2lem2 ((a v (a' ^ b')) ->2 a) = (a v b)
Proof of Theorem ud2lem2
StepHypRef Expression
1 df-i2 45 . 2 ((a v (a' ^ b')) ->2 a) = (a v ((a v (a' ^ b'))' ^ a'))
2 oran 87 . . . . . . 7 ((a v (a' ^ b')) v a) = ((a v (a' ^ b'))' ^ a')'
32con2 67 . . . . . 6 ((a v (a' ^ b')) v a)' = ((a v (a' ^ b'))' ^ a')
43ax-r1 35 . . . . 5 ((a v (a' ^ b'))' ^ a') = ((a v (a' ^ b')) v a)'
5 oran 87 . . . . . . . . . . . . 13 (a v b) = (a' ^ b')'
65con2 67 . . . . . . . . . . . 12 (a v b)' = (a' ^ b')
76ax-r1 35 . . . . . . . . . . 11 (a' ^ b') = (a v b)'
87lor 70 . . . . . . . . . 10 (a v (a' ^ b')) = (a v (a v b)')
9 anor2 89 . . . . . . . . . . . 12 (a' ^ (a v b)) = (a v (a v b)')'
109ax-r1 35 . . . . . . . . . . 11 (a v (a v b)')' = (a' ^ (a v b))
1110con3 68 . . . . . . . . . 10 (a v (a v b)') = (a' ^ (a v b))'
128, 11ax-r2 36 . . . . . . . . 9 (a v (a' ^ b')) = (a' ^ (a v b))'
1312con2 67 . . . . . . . 8 (a v (a' ^ b'))' = (a' ^ (a v b))
1413ran 78 . . . . . . 7 ((a v (a' ^ b'))' ^ a') = ((a' ^ (a v b)) ^ a')
15 an32 83 . . . . . . . 8 ((a' ^ (a v b)) ^ a') = ((a' ^ a') ^ (a v b))
16 anidm 111 . . . . . . . . 9 (a' ^ a') = a'
1716ran 78 . . . . . . . 8 ((a' ^ a') ^ (a v b)) = (a' ^ (a v b))
1815, 17ax-r2 36 . . . . . . 7 ((a' ^ (a v b)) ^ a') = (a' ^ (a v b))
1914, 18ax-r2 36 . . . . . 6 ((a v (a' ^ b'))' ^ a') = (a' ^ (a v b))
203, 19ax-r2 36 . . . . 5 ((a v (a' ^ b')) v a)' = (a' ^ (a v b))
214, 20ax-r2 36 . . . 4 ((a v (a' ^ b'))' ^ a') = (a' ^ (a v b))
2221lor 70 . . 3 (a v ((a v (a' ^ b'))' ^ a')) = (a v (a' ^ (a v b)))
23 oml 445 . . 3 (a v (a' ^ (a v b))) = (a v b)
2422, 23ax-r2 36 . 2 (a v ((a v (a' ^ b'))' ^ a')) = (a v b)
251, 24ax-r2 36 1 ((a v (a' ^ b')) ->2 a) = (a v b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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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