Theorem u3lemax5 779

Description: Possible axiom for Kalmbach implication system.

Assertion

Ref	Expression
u3lemax5	((a ->3 b) ->3 ((a ->3 b) ->3 ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))))) = 1

Proof of Theorem u3lemax5

Step	Hyp	Ref	Expression
1		lem4 493	. 2 ((a ->3 b) ->3 ((a ->3 b) ->3 ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))))) = ((a ->3 b)_|_ v ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))))
2		lem4 493	. . . . 5 ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))) = ((b ->3 a)_|_ v ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))
3		lem4 493	. . . . . . 7 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))) = ((b ->3 c)_|_ v ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))
4		lem4 493	. . . . . . . . 9 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))) = ((c ->3 b)_|_ v (a ->3 c))
5	4	lor 66	. . . . . . . 8 ((b ->3 c)_|_ v ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c)))) = ((b ->3 c)_|_ v ((c ->3 b)_|_ v (a ->3 c)))
6		ax-a3 31	. . . . . . . . . 10 (((b ->3 c)_|_ v (c ->3 b)_|_) v (a ->3 c)) = ((b ->3 c)_|_ v ((c ->3 b)_|_ v (a ->3 c)))
7	6	ax-r1 34	. . . . . . . . 9 ((b ->3 c)_|_ v ((c ->3 b)_|_ v (a ->3 c))) = (((b ->3 c)_|_ v (c ->3 b)_|_) v (a ->3 c))
8		oran3 85	. . . . . . . . . . 11 ((b ->3 c)_|_ v (c ->3 b)_|_) = ((b ->3 c) ^ (c ->3 b))_|_
9		u3lembi 705	. . . . . . . . . . . 12 ((b ->3 c) ^ (c ->3 b)) = (b == c)
10	9	ax-r4 36	. . . . . . . . . . 11 ((b ->3 c) ^ (c ->3 b))_|_ = (b == c)_|_
11	8, 10	ax-r2 35	. . . . . . . . . 10 ((b ->3 c)_|_ v (c ->3 b)_|_) = (b == c)_|_
12	11	ax-r5 37	. . . . . . . . 9 (((b ->3 c)_|_ v (c ->3 b)_|_) v (a ->3 c)) = ((b == c)_|_ v (a ->3 c))
13	7, 12	ax-r2 35	. . . . . . . 8 ((b ->3 c)_|_ v ((c ->3 b)_|_ v (a ->3 c))) = ((b == c)_|_ v (a ->3 c))
14	5, 13	ax-r2 35	. . . . . . 7 ((b ->3 c)_|_ v ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c)))) = ((b == c)_|_ v (a ->3 c))
15	3, 14	ax-r2 35	. . . . . 6 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))) = ((b == c)_|_ v (a ->3 c))
16	15	lor 66	. . . . 5 ((b ->3 a)_|_ v ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c)))))) = ((b ->3 a)_|_ v ((b == c)_|_ v (a ->3 c)))
17	2, 16	ax-r2 35	. . . 4 ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))) = ((b ->3 a)_|_ v ((b == c)_|_ v (a ->3 c)))
18	17	lor 66	. . 3 ((a ->3 b)_|_ v ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c)))))))) = ((a ->3 b)_|_ v ((b ->3 a)_|_ v ((b == c)_|_ v (a ->3 c))))
19		ax-a3 31	. . . . 5 (((a ->3 b)_|_ v (b ->3 a)_|_) v ((b == c)_|_ v (a ->3 c))) = ((a ->3 b)_|_ v ((b ->3 a)_|_ v ((b == c)_|_ v (a ->3 c))))
20	19	ax-r1 34	. . . 4 ((a ->3 b)_|_ v ((b ->3 a)_|_ v ((b == c)_|_ v (a ->3 c)))) = (((a ->3 b)_|_ v (b ->3 a)_|_) v ((b == c)_|_ v (a ->3 c)))
21		oran3 85	. . . . . . 7 ((a ->3 b)_|_ v (b ->3 a)_|_) = ((a ->3 b) ^ (b ->3 a))_|_
22		u3lembi 705	. . . . . . . 8 ((a ->3 b) ^ (b ->3 a)) = (a == b)
23	22	ax-r4 36	. . . . . . 7 ((a ->3 b) ^ (b ->3 a))_|_ = (a == b)_|_
24	21, 23	ax-r2 35	. . . . . 6 ((a ->3 b)_|_ v (b ->3 a)_|_) = (a == b)_|_
25	24	ax-r5 37	. . . . 5 (((a ->3 b)_|_ v (b ->3 a)_|_) v ((b == c)_|_ v (a ->3 c))) = ((a == b)_|_ v ((b == c)_|_ v (a ->3 c)))
26		le1 138	. . . . . 6 ((a == b)_|_ v ((b == c)_|_ v (a ->3 c))) =< 1
27		ska2 414	. . . . . . . 8 ((a == b)_|_ v ((b == c)_|_ v (a == c))) = 1
28	27	ax-r1 34	. . . . . . 7 1 = ((a == b)_|_ v ((b == c)_|_ v (a == c)))
29		u3lembi 705	. . . . . . . . . . 11 ((a ->3 c) ^ (c ->3 a)) = (a == c)
30	29	ax-r1 34	. . . . . . . . . 10 (a == c) = ((a ->3 c) ^ (c ->3 a))
31		lea 152	. . . . . . . . . 10 ((a ->3 c) ^ (c ->3 a)) =< (a ->3 c)
32	30, 31	bltr 130	. . . . . . . . 9 (a == c) =< (a ->3 c)
33	32	lelor 158	. . . . . . . 8 ((b == c)_|_ v (a == c)) =< ((b == c)_|_ v (a ->3 c))
34	33	lelor 158	. . . . . . 7 ((a == b)_|_ v ((b == c)_|_ v (a == c))) =< ((a == b)_|_ v ((b == c)_|_ v (a ->3 c)))
35	28, 34	bltr 130	. . . . . 6 1 =< ((a == b)_|_ v ((b == c)_|_ v (a ->3 c)))
36	26, 35	lebi 137	. . . . 5 ((a == b)_|_ v ((b == c)_|_ v (a ->3 c))) = 1
37	25, 36	ax-r2 35	. . . 4 (((a ->3 b)_|_ v (b ->3 a)_|_) v ((b == c)_|_ v (a ->3 c))) = 1
38	20, 37	ax-r2 35	. . 3 ((a ->3 b)_|_ v ((b ->3 a)_|_ v ((b == c)_|_ v (a ->3 c)))) = 1
39	18, 38	ax-r2 35	. 2 ((a ->3 b)_|_ v ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c)))))))) = 1
40	1, 39	ax-r2 35	1 ((a ->3 b) ->3 ((a ->3 b) ->3 ((b ->3 a) ->3 ((b ->3 a) ->3 ((b ->3 c) ->3 ((b ->3 c) ->3 ((c ->3 b) ->3 ((c ->3 b) ->3 (a ->3 c))))))))) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9   ->3 wi3 15

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-wom 343  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-i3 45  df-le 121  df-le1 122  df-le2 123  df-c1 124  df-c2 125  df-cmtr 126

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