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Theorem u2lem1 735
Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem1 ((a ->2 b) ->2 a) = a
Proof of Theorem u2lem1
StepHypRef Expression
1 df-i2 45 . 2 ((a ->2 b) ->2 a) = (a v ((a ->2 b)' ^ a'))
2 ud2lem0c 278 . . . . . 6 (a ->2 b)' = (b' ^ (a v b))
32ran 78 . . . . 5 ((a ->2 b)' ^ a') = ((b' ^ (a v b)) ^ a')
4 an32 83 . . . . . 6 ((b' ^ (a v b)) ^ a') = ((b' ^ a') ^ (a v b))
5 ax-a2 31 . . . . . . . . 9 (a v b) = (b v a)
6 oran 87 . . . . . . . . 9 (b v a) = (b' ^ a')'
75, 6ax-r2 36 . . . . . . . 8 (a v b) = (b' ^ a')'
87lan 77 . . . . . . 7 ((b' ^ a') ^ (a v b)) = ((b' ^ a') ^ (b' ^ a')')
9 dff 101 . . . . . . . 8 0 = ((b' ^ a') ^ (b' ^ a')')
109ax-r1 35 . . . . . . 7 ((b' ^ a') ^ (b' ^ a')') = 0
118, 10ax-r2 36 . . . . . 6 ((b' ^ a') ^ (a v b)) = 0
124, 11ax-r2 36 . . . . 5 ((b' ^ (a v b)) ^ a') = 0
133, 12ax-r2 36 . . . 4 ((a ->2 b)' ^ a') = 0
1413lor 70 . . 3 (a v ((a ->2 b)' ^ a')) = (a v 0)
15 or0 102 . . 3 (a v 0) = a
1614, 15ax-r2 36 . 2 (a v ((a ->2 b)' ^ a')) = a
171, 16ax-r2 36 1 ((a ->2 b) ->2 a) = a
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-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45
This theorem is referenced by:  u2lem1n  740
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