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Theorem u2lemaa 601
Description: Lemma for Dishkant implication study.
Assertion
Ref Expression
u2lemaa ((a ->2 b) ^ a) = (a ^ b)
Proof of Theorem u2lemaa
StepHypRef Expression
1 df-i2 45 . . 3 (a ->2 b) = (b v (a' ^ b'))
21ran 78 . 2 ((a ->2 b) ^ a) = ((b v (a' ^ b')) ^ a)
3 ax-a2 31 . . . 4 (b v (a' ^ b')) = ((a' ^ b') v b)
43ran 78 . . 3 ((b v (a' ^ b')) ^ a) = (((a' ^ b') v b) ^ a)
5 coman1 185 . . . . . 6 (a' ^ b') C a'
65comcom7 460 . . . . 5 (a' ^ b') C a
7 coman2 186 . . . . . 6 (a' ^ b') C b'
87comcom7 460 . . . . 5 (a' ^ b') C b
96, 8fh2r 474 . . . 4 (((a' ^ b') v b) ^ a) = (((a' ^ b') ^ a) v (b ^ a))
10 ax-a2 31 . . . . 5 (((a' ^ b') ^ a) v (b ^ a)) = ((b ^ a) v ((a' ^ b') ^ a))
11 ancom 74 . . . . . . 7 (b ^ a) = (a ^ b)
12 ancom 74 . . . . . . . 8 ((a' ^ b') ^ a) = (a ^ (a' ^ b'))
13 anass 76 . . . . . . . . . 10 ((a ^ a') ^ b') = (a ^ (a' ^ b'))
1413ax-r1 35 . . . . . . . . 9 (a ^ (a' ^ b')) = ((a ^ a') ^ b')
15 ancom 74 . . . . . . . . . 10 ((a ^ a') ^ b') = (b' ^ (a ^ a'))
16 dff 101 . . . . . . . . . . . . 13 0 = (a ^ a')
1716ax-r1 35 . . . . . . . . . . . 12 (a ^ a') = 0
1817lan 77 . . . . . . . . . . 11 (b' ^ (a ^ a')) = (b' ^ 0)
19 an0 108 . . . . . . . . . . 11 (b' ^ 0) = 0
2018, 19ax-r2 36 . . . . . . . . . 10 (b' ^ (a ^ a')) = 0
2115, 20ax-r2 36 . . . . . . . . 9 ((a ^ a') ^ b') = 0
2214, 21ax-r2 36 . . . . . . . 8 (a ^ (a' ^ b')) = 0
2312, 22ax-r2 36 . . . . . . 7 ((a' ^ b') ^ a) = 0
2411, 232or 72 . . . . . 6 ((b ^ a) v ((a' ^ b') ^ a)) = ((a ^ b) v 0)
25 or0 102 . . . . . 6 ((a ^ b) v 0) = (a ^ b)
2624, 25ax-r2 36 . . . . 5 ((b ^ a) v ((a' ^ b') ^ a)) = (a ^ b)
2710, 26ax-r2 36 . . . 4 (((a' ^ b') ^ a) v (b ^ a)) = (a ^ b)
289, 27ax-r2 36 . . 3 (((a' ^ b') v b) ^ a) = (a ^ b)
294, 28ax-r2 36 . 2 ((b v (a' ^ b')) ^ a) = (a ^ b)
302, 29ax-r2 36 1 ((a ->2 b) ^ a) = (a ^ b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->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:  u2lemnona  666
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