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Theorem u4lemc1 683
Description: Commutation theorem for non-tollens implication.
Assertion
Ref Expression
u4lemc1 b C (a ->4 b)
Proof of Theorem u4lemc1
StepHypRef Expression
1 comanr2 465 . . . 4 b C (a ^ b)
2 comanr2 465 . . . 4 b C (a' ^ b)
31, 2com2or 483 . . 3 b C ((a ^ b) v (a' ^ b))
4 comorr2 463 . . . 4 b C (a' v b)
5 comid 187 . . . . 5 b C b
65comcom2 183 . . . 4 b C b'
74, 6com2an 484 . . 3 b C ((a' v b) ^ b')
83, 7com2or 483 . 2 b C (((a ^ b) v (a' ^ b)) v ((a' v b) ^ b'))
9 df-i4 47 . . 3 (a ->4 b) = (((a ^ b) v (a' ^ b)) v ((a' v b) ^ b'))
109ax-r1 35 . 2 (((a ^ b) v (a' ^ b)) v ((a' v b) ^ b')) = (a ->4 b)
118, 10cbtr 182 1 b C (a ->4 b)
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7   ->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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u4lemc3  694  u4lem2  747  u4lem3  752  u24lem  770
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