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Theorem i5con 272
Description: Converse of ->5.
Assertion
Ref Expression
i5con (a ->5 b) = (b' ->5 a')
Proof of Theorem i5con
StepHypRef Expression
1 ancom 74 . . . 4 (a' ^ b') = (b' ^ a')
2 ax-a2 31 . . . . 5 ((a ^ b) v (a' ^ b)) = ((a' ^ b) v (a ^ b))
3 ancom 74 . . . . . . 7 (a' ^ b) = (b ^ a')
4 ax-a1 30 . . . . . . . 8 b = b''
54ran 78 . . . . . . 7 (b ^ a') = (b'' ^ a')
63, 5ax-r2 36 . . . . . 6 (a' ^ b) = (b'' ^ a')
7 ancom 74 . . . . . . 7 (a ^ b) = (b ^ a)
8 ax-a1 30 . . . . . . . 8 a = a''
94, 82an 79 . . . . . . 7 (b ^ a) = (b'' ^ a'')
107, 9ax-r2 36 . . . . . 6 (a ^ b) = (b'' ^ a'')
116, 102or 72 . . . . 5 ((a' ^ b) v (a ^ b)) = ((b'' ^ a') v (b'' ^ a''))
122, 11ax-r2 36 . . . 4 ((a ^ b) v (a' ^ b)) = ((b'' ^ a') v (b'' ^ a''))
131, 122or 72 . . 3 ((a' ^ b') v ((a ^ b) v (a' ^ b))) = ((b' ^ a') v ((b'' ^ a') v (b'' ^ a'')))
14 ax-a2 31 . . 3 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = ((a' ^ b') v ((a ^ b) v (a' ^ b)))
15 ax-a3 32 . . 3 (((b' ^ a') v (b'' ^ a')) v (b'' ^ a'')) = ((b' ^ a') v ((b'' ^ a') v (b'' ^ a'')))
1613, 14, 153tr1 63 . 2 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (((b' ^ a') v (b'' ^ a')) v (b'' ^ a''))
17 df-i5 48 . 2 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
18 df-i5 48 . 2 (b' ->5 a') = (((b' ^ a') v (b'' ^ a')) v (b'' ^ a''))
1916, 17, 183tr1 63 1 (a ->5 b) = (b' ->5 a')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->5 wi5 16
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i5 48
This theorem is referenced by:  nom45  330
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