Theorem 3vroa 813

Description: OA-like inference rule (requires OM only).

Hypothesis

Ref	Expression
3vroa.1	((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = 1

Assertion

Ref	Expression
3vroa	(a ->2 c) = 1

Proof of Theorem 3vroa

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a ->2 c) = (c v (a_|_ ^ c_|_))
2		or12 73	. . 3 (c v ((a_|_ ^ c_|_) v (a_|_ ^ c_|_))) = ((a_|_ ^ c_|_) v (c v (a_|_ ^ c_|_)))
3		oridm 102	. . . 4 ((a_|_ ^ c_|_) v (a_|_ ^ c_|_)) = (a_|_ ^ c_|_)
4	3	lor 66	. . 3 (c v ((a_|_ ^ c_|_) v (a_|_ ^ c_|_))) = (c v (a_|_ ^ c_|_))
5		le1 138	. . . . . . . . . 10 (a ->2 b) =< 1
6		3vroa.1	. . . . . . . . . . . 12 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) = 1
7	6	ax-r1 34	. . . . . . . . . . 11 1 = ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))))
8		lea 152	. . . . . . . . . . 11 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 b)
9	7, 8	bltr 130	. . . . . . . . . 10 1 =< (a ->2 b)
10	5, 9	lebi 137	. . . . . . . . 9 (a ->2 b) = 1
11	10	ran 71	. . . . . . . 8 ((a ->2 b) ^ (a ->2 c)) = (1 ^ (a ->2 c))
12		ancom 68	. . . . . . . 8 (1 ^ (a ->2 c)) = ((a ->2 c) ^ 1)
13	11, 12	ax-r2 35	. . . . . . 7 ((a ->2 b) ^ (a ->2 c)) = ((a ->2 c) ^ 1)
14		an1 98	. . . . . . 7 ((a ->2 c) ^ 1) = (a ->2 c)
15	13, 14, 1	3tr 62	. . . . . 6 ((a ->2 b) ^ (a ->2 c)) = (c v (a_|_ ^ c_|_))
16	15	lor 66	. . . . 5 ((a_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))) = ((a_|_ ^ c_|_) v (c v (a_|_ ^ c_|_)))
17	16	ax-r1 34	. . . 4 ((a_|_ ^ c_|_) v (c v (a_|_ ^ c_|_))) = ((a_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
18		le1 138	. . . . 5 ((a_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))) =< 1
19		lear 153	. . . . . . . 8 ((a ->2 b) ^ ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
20		df-i0 42	. . . . . . . . 9 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
21		anor3 82	. . . . . . . . . . 11 (b_|_ ^ c_|_) = (b v c)_|_
22	21	ax-r5 37	. . . . . . . . . 10 ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
23	22	ax-r1 34	. . . . . . . . 9 ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) = ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
24	20, 23	ax-r2 35	. . . . . . . 8 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
25	19, 6, 24	le3tr2 133	. . . . . . 7 1 =< ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
26		le1 138	. . . . . . 7 ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))) =< 1
27	25, 26	lebi 137	. . . . . 6 1 = ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
28	10	u2lemle2 698	. . . . . . . . 9 a =< b
29	28	lecon 146	. . . . . . . 8 b_|_ =< a_|_
30	29	leran 145	. . . . . . 7 (b_|_ ^ c_|_) =< (a_|_ ^ c_|_)
31	30	leror 144	. . . . . 6 ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))) =< ((a_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
32	27, 31	bltr 130	. . . . 5 1 =< ((a_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))
33	18, 32	lebi 137	. . . 4 ((a_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))) = 1
34	17, 33	ax-r2 35	. . 3 ((a_|_ ^ c_|_) v (c v (a_|_ ^ c_|_))) = 1
35	2, 4, 34	3tr2 61	. 2 (c v (a_|_ ^ c_|_)) = 1
36	1, 35	ax-r2 35	1 (a ->2 c) = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->0 wi0 12   ->2 wi2 14

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-i0 42  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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