Theorem ud2 582

Description: Unified disjunction for Dishkant implication.

Assertion

Ref	Expression
ud2	(a v b) = ((a ->2 b) ->2 (((a ->2 b) ->2 (b ->2 a)) ->2 a))

Proof of Theorem ud2

Step	Hyp	Ref	Expression
1		ud2lem1 549	. . . . . 6 ((a ->2 b) ->2 (b ->2 a)) = (a v (a' ^ b'))
2	1	ud2lem0b 253	. . . . 5 (((a ->2 b) ->2 (b ->2 a)) ->2 a) = ((a v (a' ^ b')) ->2 a)
3		ud2lem2 550	. . . . 5 ((a v (a' ^ b')) ->2 a) = (a v b)
4	2, 3	ax-r2 36	. . . 4 (((a ->2 b) ->2 (b ->2 a)) ->2 a) = (a v b)
5	4	ud2lem0a 252	. . 3 ((a ->2 b) ->2 (((a ->2 b) ->2 (b ->2 a)) ->2 a)) = ((a ->2 b) ->2 (a v b))
6		ud2lem3 551	. . 3 ((a ->2 b) ->2 (a v b)) = (a v b)
7	5, 6	ax-r2 36	. 2 ((a ->2 b) ->2 (((a ->2 b) ->2 (b ->2 a)) ->2 a)) = (a v b)
8	7	ax-r1 35	1 (a v b) = ((a ->2 b) ->2 (((a ->2 b) ->2 (b ->2 a)) ->2 a))

Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13

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-i2 45  df-le1 124  df-le2 125  df-c1 126  df-c2 127

Copyright terms: Public domain