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Theorem dfi4b 500
Description: Alternate non-tollens conditional.
Assertion
Ref Expression
dfi4b (a ->4 b) = ((a' v b) ^ ((b' v (b ^ a')) v (b ^ a)))
Proof of Theorem dfi4b
StepHypRef Expression
1 i4i3 271 . 2 (a ->4 b) = (b' ->3 a')
2 dfi3b 499 . . 3 (b' ->3 a') = ((b'' v a') ^ ((b' v (b'' ^ a'')) v (b'' ^ a')))
3 ax-a2 31 . . . . . 6 (a' v b) = (b v a')
4 ax-a1 30 . . . . . . 7 b = b''
54ax-r5 38 . . . . . 6 (b v a') = (b'' v a')
63, 5ax-r2 36 . . . . 5 (a' v b) = (b'' v a')
74ran 78 . . . . . . . 8 (b ^ a') = (b'' ^ a')
87lor 70 . . . . . . 7 (b' v (b ^ a')) = (b' v (b'' ^ a'))
9 ax-a1 30 . . . . . . . 8 a = a''
104, 92an 79 . . . . . . 7 (b ^ a) = (b'' ^ a'')
118, 102or 72 . . . . . 6 ((b' v (b ^ a')) v (b ^ a)) = ((b' v (b'' ^ a')) v (b'' ^ a''))
12 or32 82 . . . . . 6 ((b' v (b'' ^ a')) v (b'' ^ a'')) = ((b' v (b'' ^ a'')) v (b'' ^ a'))
1311, 12ax-r2 36 . . . . 5 ((b' v (b ^ a')) v (b ^ a)) = ((b' v (b'' ^ a'')) v (b'' ^ a'))
146, 132an 79 . . . 4 ((a' v b) ^ ((b' v (b ^ a')) v (b ^ a))) = ((b'' v a') ^ ((b' v (b'' ^ a'')) v (b'' ^ a')))
1514ax-r1 35 . . 3 ((b'' v a') ^ ((b' v (b'' ^ a'')) v (b'' ^ a'))) = ((a' v b) ^ ((b' v (b ^ a')) v (b ^ a)))
162, 15ax-r2 36 . 2 (b' ->3 a') = ((a' v b) ^ ((b' v (b ^ a')) v (b ^ a)))
171, 16ax-r2 36 1 (a ->4 b) = ((a' v b) ^ ((b' v (b ^ a')) v (b ^ 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-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-i3 46  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  negantlem10  861
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