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Theorem i3i4 270
Description: Correspondence between Kalmbach and non-tollens conditionals.
Assertion
Ref Expression
i3i4 (a ->3 b) = (b' ->4 a')
Proof of Theorem i3i4
StepHypRef Expression
1 ax-a2 31 . . . 4 ((a' ^ b) v (a' ^ b')) = ((a' ^ b') v (a' ^ b))
2 ancom 74 . . . . 5 (a' ^ b') = (b' ^ a')
3 ancom 74 . . . . . 6 (a' ^ b) = (b ^ a')
4 ax-a1 30 . . . . . . 7 b = b''
54ran 78 . . . . . 6 (b ^ a') = (b'' ^ a')
63, 5ax-r2 36 . . . . 5 (a' ^ b) = (b'' ^ a')
72, 62or 72 . . . 4 ((a' ^ b') v (a' ^ b)) = ((b' ^ a') v (b'' ^ a'))
81, 7ax-r2 36 . . 3 ((a' ^ b) v (a' ^ b')) = ((b' ^ a') v (b'' ^ a'))
9 ancom 74 . . . 4 (a ^ (a' v b)) = ((a' v b) ^ a)
10 ax-a2 31 . . . . . 6 (a' v b) = (b v a')
114ax-r5 38 . . . . . 6 (b v a') = (b'' v a')
1210, 11ax-r2 36 . . . . 5 (a' v b) = (b'' v a')
13 ax-a1 30 . . . . 5 a = a''
1412, 132an 79 . . . 4 ((a' v b) ^ a) = ((b'' v a') ^ a'')
159, 14ax-r2 36 . . 3 (a ^ (a' v b)) = ((b'' v a') ^ a'')
168, 152or 72 . 2 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) = (((b' ^ a') v (b'' ^ a')) v ((b'' v a') ^ a''))
17 df-i3 46 . 2 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
18 df-i4 47 . 2 (b' ->4 a') = (((b' ^ a') v (b'' ^ a')) v ((b'' v a') ^ a''))
1916, 17, 183tr1 63 1 (a ->3 b) = (b' ->4 a')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14   ->4 wi4 15
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i3 46  df-i4 47
This theorem is referenced by:  i4i3  271  nom43  328
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