Theorem 3vth1 786

Description: A 3-variable theorem. Equivalent to OML.

Assertion

Ref	Expression
3vth1	((a ->2 b) ^ (b v c)_|_) =< (a ->2 c)

Proof of Theorem 3vth1

Step	Hyp	Ref	Expression
1		anor3 82	. . . . . . 7 (b_|_ ^ c_|_) = (b v c)_|_
2	1	lan 70	. . . . . 6 ((b v (b_|_ ^ a_|_)) ^ (b_|_ ^ c_|_)) = ((b v (b_|_ ^ a_|_)) ^ (b v c)_|_)
3	2	ax-r1 34	. . . . 5 ((b v (b_|_ ^ a_|_)) ^ (b v c)_|_) = ((b v (b_|_ ^ a_|_)) ^ (b_|_ ^ c_|_))
4		anass 69	. . . . . 6 (((b v (b_|_ ^ a_|_)) ^ b_|_) ^ c_|_) = ((b v (b_|_ ^ a_|_)) ^ (b_|_ ^ c_|_))
5	4	ax-r1 34	. . . . 5 ((b v (b_|_ ^ a_|_)) ^ (b_|_ ^ c_|_)) = (((b v (b_|_ ^ a_|_)) ^ b_|_) ^ c_|_)
6	3, 5	ax-r2 35	. . . 4 ((b v (b_|_ ^ a_|_)) ^ (b v c)_|_) = (((b v (b_|_ ^ a_|_)) ^ b_|_) ^ c_|_)
7		ancom 68	. . . . . . 7 ((b v (b_|_ ^ a_|_)) ^ b_|_) = (b_|_ ^ (b v (b_|_ ^ a_|_)))
8		omlan 430	. . . . . . 7 (b_|_ ^ (b v (b_|_ ^ a_|_))) = (b_|_ ^ a_|_)
9	7, 8	ax-r2 35	. . . . . 6 ((b v (b_|_ ^ a_|_)) ^ b_|_) = (b_|_ ^ a_|_)
10		lear 153	. . . . . 6 (b_|_ ^ a_|_) =< a_|_
11	9, 10	bltr 130	. . . . 5 ((b v (b_|_ ^ a_|_)) ^ b_|_) =< a_|_
12	11	leran 145	. . . 4 (((b v (b_|_ ^ a_|_)) ^ b_|_) ^ c_|_) =< (a_|_ ^ c_|_)
13	6, 12	bltr 130	. . 3 ((b v (b_|_ ^ a_|_)) ^ (b v c)_|_) =< (a_|_ ^ c_|_)
14		leor 151	. . 3 (a_|_ ^ c_|_) =< (c v (a_|_ ^ c_|_))
15	13, 14	letr 129	. 2 ((b v (b_|_ ^ a_|_)) ^ (b v c)_|_) =< (c v (a_|_ ^ c_|_))
16		df-i2 44	. . . 4 (a ->2 b) = (b v (a_|_ ^ b_|_))
17		ancom 68	. . . . 5 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
18	17	lor 66	. . . 4 (b v (a_|_ ^ b_|_)) = (b v (b_|_ ^ a_|_))
19	16, 18	ax-r2 35	. . 3 (a ->2 b) = (b v (b_|_ ^ a_|_))
20	19	ran 71	. 2 ((a ->2 b) ^ (b v c)_|_) = ((b v (b_|_ ^ a_|_)) ^ (b v c)_|_)
21		df-i2 44	. 2 (a ->2 c) = (c v (a_|_ ^ c_|_))
22	15, 20, 21	le3tr1 132	1 ((a ->2 b) ^ (b v c)_|_) =< (a ->2 c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  3vth2 787  3vth3 788

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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-i2 44  df-le1 122  df-le2 123

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