Theorem ud5 581

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 571	. . . . . 6 ((a ->5 b) ->5 (b ->5 a)) = (a v b_|_)
2	1	ud5lem0b 257	. . . . 5 (((a ->5 b) ->5 (b ->5 a)) ->5 a) = ((a v b_|_) ->5 a)
3		ud5lem2 572	. . . . 5 ((a v b_|_) ->5 a) = (a v (a_|_ ^ b))
4	2, 3	ax-r2 35	. . . 4 (((a ->5 b) ->5 (b ->5 a)) ->5 a) = (a v (a_|_ ^ b))
5	4	ud5lem0a 256	. . 3 ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a)) = ((a ->5 b) ->5 (a v (a_|_ ^ b)))
6		ud5lem3 576	. . 3 ((a ->5 b) ->5 (a v (a_|_ ^ b))) = (a v b)
7	5, 6	ax-r2 35	. 2 ((a ->5 b) ->5 (((a ->5 b) ->5 (b ->5 a)) ->5 a)) = (a v b)
8	7	ax-r1 34	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 17

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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