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Theorem ud1lem2 561
Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud1lem2 ((a v (a' ^ b')) ->1 a) = (a v b)
Proof of Theorem ud1lem2
StepHypRef Expression
1 df-i1 44 . 2 ((a v (a' ^ b')) ->1 a) = ((a v (a' ^ b'))' v ((a v (a' ^ b')) ^ a))
2 comid 187 . . . 4 (a v (a' ^ b')) C (a v (a' ^ b'))
32comcom3 454 . . 3 (a v (a' ^ b'))' C (a v (a' ^ b'))
4 comor1 461 . . . 4 (a v (a' ^ b')) C a
54comcom3 454 . . 3 (a v (a' ^ b'))' C a
63, 5fh3 471 . 2 ((a v (a' ^ b'))' v ((a v (a' ^ b')) ^ a)) = (((a v (a' ^ b'))' v (a v (a' ^ b'))) ^ ((a v (a' ^ b'))' v a))
7 ancom 74 . . 3 (((a v (a' ^ b'))' v (a v (a' ^ b'))) ^ ((a v (a' ^ b'))' v a)) = (((a v (a' ^ b'))' v a) ^ ((a v (a' ^ b'))' v (a v (a' ^ b'))))
8 ax-a2 31 . . . . 5 ((a v (a' ^ b'))' v (a v (a' ^ b'))) = ((a v (a' ^ b')) v (a v (a' ^ b'))')
9 df-t 41 . . . . . 6 1 = ((a v (a' ^ b')) v (a v (a' ^ b'))')
109ax-r1 35 . . . . 5 ((a v (a' ^ b')) v (a v (a' ^ b'))') = 1
118, 10ax-r2 36 . . . 4 ((a v (a' ^ b'))' v (a v (a' ^ b'))) = 1
1211lan 77 . . 3 (((a v (a' ^ b'))' v a) ^ ((a v (a' ^ b'))' v (a v (a' ^ b')))) = (((a v (a' ^ b'))' v a) ^ 1)
13 an1 106 . . . 4 (((a v (a' ^ b'))' v a) ^ 1) = ((a v (a' ^ b'))' v a)
14 oran 87 . . . . . . 7 (a v (a' ^ b')) = (a' ^ (a' ^ b')')'
15 oran 87 . . . . . . . . . 10 (a v b) = (a' ^ b')'
1615ax-r1 35 . . . . . . . . 9 (a' ^ b')' = (a v b)
1716lan 77 . . . . . . . 8 (a' ^ (a' ^ b')') = (a' ^ (a v b))
1817ax-r4 37 . . . . . . 7 (a' ^ (a' ^ b')')' = (a' ^ (a v b))'
1914, 18ax-r2 36 . . . . . 6 (a v (a' ^ b')) = (a' ^ (a v b))'
2019con2 67 . . . . 5 (a v (a' ^ b'))' = (a' ^ (a v b))
2120ax-r5 38 . . . 4 ((a v (a' ^ b'))' v a) = ((a' ^ (a v b)) v a)
22 ax-a2 31 . . . . 5 ((a' ^ (a v b)) v a) = (a v (a' ^ (a v b)))
23 oml 445 . . . . 5 (a v (a' ^ (a v b))) = (a v b)
2422, 23ax-r2 36 . . . 4 ((a' ^ (a v b)) v a) = (a v b)
2513, 21, 243tr 65 . . 3 (((a v (a' ^ b'))' v a) ^ 1) = (a v b)
267, 12, 253tr 65 . 2 (((a v (a' ^ b'))' v (a v (a' ^ b'))) ^ ((a v (a' ^ b'))' v a)) = (a v b)
271, 6, 263tr 65 1 ((a v (a' ^ b')) ->1 a) = (a v b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  ud1  595
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