Theorem ud1 581

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

Step	Hyp	Ref	Expression
1		ud1lem1 546	. . . . . 6 ((a ->1 b) ->1 (b ->1 a)) = (a v (a' ^ b'))
2	1	ud1lem0b 250	. . . . 5 (((a ->1 b) ->1 (b ->1 a)) ->1 a) = ((a v (a' ^ b')) ->1 a)
3		ud1lem2 547	. . . . 5 ((a v (a' ^ b')) ->1 a) = (a v b)
4	2, 3	ax-r2 36	. . . 4 (((a ->1 b) ->1 (b ->1 a)) ->1 a) = (a v b)
5	4	ud1lem0a 249	. . 3 ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a)) = ((a ->1 b) ->1 (a v b))
6		ud1lem3 548	. . 3 ((a ->1 b) ->1 (a v b)) = (a v b)
7	5, 6	ax-r2 36	. 2 ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a)) = (a v b)
8	7	ax-r1 35	1 (a v b) = ((a ->1 b) ->1 (((a ->1 b) ->1 (b ->1 a)) ->1 a))

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 425

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 124  df-le2 125  df-c1 126  df-c2 127

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