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Theorem ud4 598
Description: Unified disjunction for non-tollens implication.
Assertion
Ref Expression
ud4 (a v b) = ((a ->4 b) ->4 (((a ->4 b) ->4 (b ->4 a)) ->4 a))

Proof of Theorem ud4
StepHypRef Expression
1 ud4lem1 581 . . . . . 6 ((a ->4 b) ->4 (b ->4 a)) = (a v (a' ^ b'))
21ud4lem0b 263 . . . . 5 (((a ->4 b) ->4 (b ->4 a)) ->4 a) = ((a v (a' ^ b')) ->4 a)
3 ud4lem2 582 . . . . 5 ((a v (a' ^ b')) ->4 a) = (a v b)
42, 3ax-r2 36 . . . 4 (((a ->4 b) ->4 (b ->4 a)) ->4 a) = (a v b)
54ud4lem0a 262 . . 3 ((a ->4 b) ->4 (((a ->4 b) ->4 (b ->4 a)) ->4 a)) = ((a ->4 b) ->4 (a v b))
6 ud4lem3 585 . . 3 ((a ->4 b) ->4 (a v b)) = (a v b)
75, 6ax-r2 36 . 2 ((a ->4 b) ->4 (((a ->4 b) ->4 (b ->4 a)) ->4 a)) = (a v b)
87ax-r1 35 1 (a v b) = ((a ->4 b) ->4 (((a ->4 b) ->4 (b ->4 a)) ->4 a))
Colors of variables: term
Syntax hints:   = wb 1  '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
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