Theorem 3vded11 796

Description: A 3-variable theorem. Experiment with weak deduction theorem.

Hypothesis

Ref	Expression
3vded11.1	b =< (c ->1 (b ->1 a))

Assertion

Ref	Expression
3vded11	c =< (b ->1 a)

Proof of Theorem 3vded11

Step	Hyp	Ref	Expression
1		le1 138	. . 3 (c ->1 (b ->1 a)) =< 1
2		df-t 40	. . . . 5 1 = ((b v c_|_) v (b v c_|_)_|_)
3		ancom 68	. . . . . . . 8 (c ^ b_|_) = (b_|_ ^ c)
4		anor2 81	. . . . . . . 8 (b_|_ ^ c) = (b v c_|_)_|_
5	3, 4	ax-r2 35	. . . . . . 7 (c ^ b_|_) = (b v c_|_)_|_
6	5	lor 66	. . . . . 6 ((b v c_|_) v (c ^ b_|_)) = ((b v c_|_) v (b v c_|_)_|_)
7	6	ax-r1 34	. . . . 5 ((b v c_|_) v (b v c_|_)_|_) = ((b v c_|_) v (c ^ b_|_))
8		ax-a3 31	. . . . 5 ((b v c_|_) v (c ^ b_|_)) = (b v (c_|_ v (c ^ b_|_)))
9	2, 7, 8	3tr 62	. . . 4 1 = (b v (c_|_ v (c ^ b_|_)))
10		3vded11.1	. . . . 5 b =< (c ->1 (b ->1 a))
11		leo 150	. . . . . . . . 9 b_|_ =< (b_|_ v (b ^ a))
12		df-i1 43	. . . . . . . . . 10 (b ->1 a) = (b_|_ v (b ^ a))
13	12	ax-r1 34	. . . . . . . . 9 (b_|_ v (b ^ a)) = (b ->1 a)
14	11, 13	lbtr 131	. . . . . . . 8 b_|_ =< (b ->1 a)
15	14	lelan 159	. . . . . . 7 (c ^ b_|_) =< (c ^ (b ->1 a))
16	15	lelor 158	. . . . . 6 (c_|_ v (c ^ b_|_)) =< (c_|_ v (c ^ (b ->1 a)))
17		df-i1 43	. . . . . . 7 (c ->1 (b ->1 a)) = (c_|_ v (c ^ (b ->1 a)))
18	17	ax-r1 34	. . . . . 6 (c_|_ v (c ^ (b ->1 a))) = (c ->1 (b ->1 a))
19	16, 18	lbtr 131	. . . . 5 (c_|_ v (c ^ b_|_)) =< (c ->1 (b ->1 a))
20	10, 19	lel2or 162	. . . 4 (b v (c_|_ v (c ^ b_|_))) =< (c ->1 (b ->1 a))
21	9, 20	bltr 130	. . 3 1 =< (c ->1 (b ->1 a))
22	1, 21	lebi 137	. 2 (c ->1 (b ->1 a)) = 1
23	22	u1lemle2 697	1 c =< (b ->1 a)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  3vded13 798

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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