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Theorem ud2lem3 565
Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud2lem3 ((a ->2 b) ->2 (a v b)) = (a v b)
Proof of Theorem ud2lem3
StepHypRef Expression
1 df-i2 45 . 2 ((a ->2 b) ->2 (a v b)) = ((a v b) v ((a ->2 b)' ^ (a v b)'))
2 ud2lem0c 278 . . . . 5 (a ->2 b)' = (b' ^ (a v b))
32ran 78 . . . 4 ((a ->2 b)' ^ (a v b)') = ((b' ^ (a v b)) ^ (a v b)')
43lor 70 . . 3 ((a v b) v ((a ->2 b)' ^ (a v b)')) = ((a v b) v ((b' ^ (a v b)) ^ (a v b)'))
5 coman2 186 . . . . . 6 (b' ^ (a v b)) C (a v b)
65comcom 453 . . . . 5 (a v b) C (b' ^ (a v b))
7 comid 187 . . . . . 6 (a v b) C (a v b)
87comcom2 183 . . . . 5 (a v b) C (a v b)'
96, 8fh3 471 . . . 4 ((a v b) v ((b' ^ (a v b)) ^ (a v b)')) = (((a v b) v (b' ^ (a v b))) ^ ((a v b) v (a v b)'))
10 ancom 74 . . . . . . 7 (b' ^ (a v b)) = ((a v b) ^ b')
1110lor 70 . . . . . 6 ((a v b) v (b' ^ (a v b))) = ((a v b) v ((a v b) ^ b'))
12 df-t 41 . . . . . . 7 1 = ((a v b) v (a v b)')
1312ax-r1 35 . . . . . 6 ((a v b) v (a v b)') = 1
1411, 132an 79 . . . . 5 (((a v b) v (b' ^ (a v b))) ^ ((a v b) v (a v b)')) = (((a v b) v ((a v b) ^ b')) ^ 1)
15 an1 106 . . . . . 6 (((a v b) v ((a v b) ^ b')) ^ 1) = ((a v b) v ((a v b) ^ b'))
16 orabs 120 . . . . . 6 ((a v b) v ((a v b) ^ b')) = (a v b)
1715, 16ax-r2 36 . . . . 5 (((a v b) v ((a v b) ^ b')) ^ 1) = (a v b)
1814, 17ax-r2 36 . . . 4 (((a v b) v (b' ^ (a v b))) ^ ((a v b) v (a v b)')) = (a v b)
199, 18ax-r2 36 . . 3 ((a v b) v ((b' ^ (a v b)) ^ (a v b)')) = (a v b)
204, 19ax-r2 36 . 2 ((a v b) v ((a ->2 b)' ^ (a v b)')) = (a v b)
211, 20ax-r2 36 1 ((a ->2 b) ->2 (a v b)) = (a v b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  ud2  596
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