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Theorem i2i1i1 800
Description: Equivalence to ->2.
Assertion
Ref Expression
i2i1i1 (a ->2 b) = ((a ->1 (a v b)) ^ ((a v b) ->1 b))
Proof of Theorem i2i1i1
StepHypRef Expression
1 an1r 107 . . 3 (1 ^ (b v (a' ^ b'))) = (b v (a' ^ b'))
21ax-r1 35 . 2 (b v (a' ^ b')) = (1 ^ (b v (a' ^ b')))
3 df-i2 45 . 2 (a ->2 b) = (b v (a' ^ b'))
4 anabs 121 . . . . . 6 (a ^ (a v b)) = a
54lor 70 . . . . 5 (a' v (a ^ (a v b))) = (a' v a)
6 ax-a2 31 . . . . 5 (a' v a) = (a v a')
75, 6ax-r2 36 . . . 4 (a' v (a ^ (a v b))) = (a v a')
8 df-i1 44 . . . 4 (a ->1 (a v b)) = (a' v (a ^ (a v b)))
9 df-t 41 . . . 4 1 = (a v a')
107, 8, 93tr1 63 . . 3 (a ->1 (a v b)) = 1
11 df-i1 44 . . . 4 ((a v b) ->1 b) = ((a v b)' v ((a v b) ^ b))
12 anor3 90 . . . . . 6 (a' ^ b') = (a v b)'
13 leor 159 . . . . . . . 8 b =< (a v b)
14 leid 148 . . . . . . . 8 b =< b
1513, 14ler2an 173 . . . . . . 7 b =< ((a v b) ^ b)
16 lear 161 . . . . . . 7 ((a v b) ^ b) =< b
1715, 16lebi 145 . . . . . 6 b = ((a v b) ^ b)
1812, 172or 72 . . . . 5 ((a' ^ b') v b) = ((a v b)' v ((a v b) ^ b))
1918ax-r1 35 . . . 4 ((a v b)' v ((a v b) ^ b)) = ((a' ^ b') v b)
20 ax-a2 31 . . . 4 ((a' ^ b') v b) = (b v (a' ^ b'))
2111, 19, 203tr 65 . . 3 ((a v b) ->1 b) = (b v (a' ^ b'))
2210, 212an 79 . 2 ((a ->1 (a v b)) ^ ((a v b) ->1 b)) = (1 ^ (b v (a' ^ b')))
232, 3, 223tr1 63 1 (a ->2 b) = ((a ->1 (a v b)) ^ ((a v b) ->1 b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12   ->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-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  mlaconj  845
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