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Theorem ud1 595
Description: Unified disjunction for Sasaki implication.
Assertion
Ref Expression
ud1 (a v b) = ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a))

Proof of Theorem ud1
StepHypRef Expression
1 ud1lem1 560 . . . . . 6 ((a ->1 b) ->1 (b ->1 a)) = (a v (a' ^ b'))
21ud1lem0b 256 . . . . 5 (((a ->1 b) ->1 (b ->1 a)) ->1 a) = ((a v (a' ^ b')) ->1 a)
3 ud1lem2 561 . . . . 5 ((a v (a' ^ b')) ->1 a) = (a v b)
42, 3ax-r2 36 . . . 4 (((a ->1 b) ->1 (b ->1 a)) ->1 a) = (a v b)
54ud1lem0a 255 . . 3 ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a)) = ((a ->1 b) ->1 (a v b))
6 ud1lem3 562 . . 3 ((a ->1 b) ->1 (a v b)) = (a v b)
75, 6ax-r2 36 . 2 ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a)) = (a v b)
87ax-r1 35 1 (a v b) = ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
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