Theorem ud3 583

Description: Unified disjunction for Kalmbach implication.

Assertion

Ref	Expression
ud3	(a v b) = ((a ->3 b) ->3 (((a ->3 b) ->3 (b ->3 a)) ->3 a))

Proof of Theorem ud3

Step	Hyp	Ref	Expression
1		ud3lem1 556	. . . . . 6 ((a ->3 b) ->3 (b ->3 a)) = (a v (a' ^ b'))
2	1	ud3lem0b 255	. . . . 5 (((a ->3 b) ->3 (b ->3 a)) ->3 a) = ((a v (a' ^ b')) ->3 a)
3		ud3lem2 557	. . . . 5 ((a v (a' ^ b')) ->3 a) = (a v b)
4	2, 3	ax-r2 36	. . . 4 (((a ->3 b) ->3 (b ->3 a)) ->3 a) = (a v b)
5	4	ud3lem0a 254	. . 3 ((a ->3 b) ->3 (((a ->3 b) ->3 (b ->3 a)) ->3 a)) = ((a ->3 b) ->3 (a v b))
6		ud3lem3 562	. . 3 ((a ->3 b) ->3 (a v b)) = (a v b)
7	5, 6	ax-r2 36	. 2 ((a ->3 b) ->3 (((a ->3 b) ->3 (b ->3 a)) ->3 a)) = (a v b)
8	7	ax-r1 35	1 (a v b) = ((a ->3 b) ->3 (((a ->3 b) ->3 (b ->3 a)) ->3 a))

Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14

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-i3 46  df-le1 124  df-le2 125  df-c1 126  df-c2 127

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