Theorem ud5 585

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

Step	Hyp	Ref	Expression
1		ud5lem1 575	. . . . . 6 ((a ->5 b) ->5 (b ->5 a)) = (a v b')
2	1	ud5lem0b 259	. . . . 5 (((a ->5 b) ->5 (b ->5 a)) ->5 a) = ((a v b') ->5 a)
3		ud5lem2 576	. . . . 5 ((a v b') ->5 a) = (a v (a' ^ b))
4	2, 3	ax-r2 36	. . . 4 (((a ->5 b) ->5 (b ->5 a)) ->5 a) = (a v (a' ^ b))
5	4	ud5lem0a 258	. . 3 ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a)) = ((a ->5 b) ->5 (a v (a' ^ b)))
6		ud5lem3 580	. . 3 ((a ->5 b) ->5 (a v (a' ^ b))) = (a v b)
7	5, 6	ax-r2 36	. 2 ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a)) = (a v b)
8	7	ax-r1 35	1 (a v b) = ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a))

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 425

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i5 48  df-le1 124  df-le2 125  df-c1 126  df-c2 127

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