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Theorem ud5 599
Description: Unified disjunction for relevance implication.
Assertion
Ref Expression
ud5 (a v b) = ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a))

Proof of Theorem ud5
StepHypRef Expression
1 ud5lem1 589 . . . . . 6 ((a ->5 b) ->5 (b ->5 a)) = (a v b')
21ud5lem0b 265 . . . . 5 (((a ->5 b) ->5 (b ->5 a)) ->5 a) = ((a v b') ->5 a)
3 ud5lem2 590 . . . . 5 ((a v b') ->5 a) = (a v (a' ^ b))
42, 3ax-r2 36 . . . 4 (((a ->5 b) ->5 (b ->5 a)) ->5 a) = (a v (a' ^ b))
54ud5lem0a 264 . . 3 ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a)) = ((a ->5 b) ->5 (a v (a' ^ b)))
6 ud5lem3 594 . . 3 ((a ->5 b) ->5 (a v (a' ^ b))) = (a v b)
75, 6ax-r2 36 . 2 ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a)) = (a v b)
87ax-r1 35 1 (a v b) = ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->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-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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133
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