Theorem u12lem 753

Description: Implication lemma.

Assertion

Ref	Expression
u12lem	((a ->1 b) v (a ->2 b)) = (a ->0 b)

Proof of Theorem u12lem

Step	Hyp	Ref	Expression
1		orordi 104	. . 3 ((a ->1 b) v (b v (a_|_ ^ b_|_))) = (((a ->1 b) v b) v ((a ->1 b) v (a_|_ ^ b_|_)))
2		u1lemob 612	. . . . 5 ((a ->1 b) v b) = (a_|_ v b)
3		df-i1 43	. . . . . . 7 (a ->1 b) = (a_|_ v (a ^ b))
4	3	ax-r5 37	. . . . . 6 ((a ->1 b) v (a_|_ ^ b_|_)) = ((a_|_ v (a ^ b)) v (a_|_ ^ b_|_))
5		or32 75	. . . . . . 7 ((a_|_ v (a ^ b)) v (a_|_ ^ b_|_)) = ((a_|_ v (a_|_ ^ b_|_)) v (a ^ b))
6		a5b 112	. . . . . . . 8 (a_|_ v (a_|_ ^ b_|_)) = a_|_
7	6	ax-r5 37	. . . . . . 7 ((a_|_ v (a_|_ ^ b_|_)) v (a ^ b)) = (a_|_ v (a ^ b))
8	5, 7	ax-r2 35	. . . . . 6 ((a_|_ v (a ^ b)) v (a_|_ ^ b_|_)) = (a_|_ v (a ^ b))
9	4, 8	ax-r2 35	. . . . 5 ((a ->1 b) v (a_|_ ^ b_|_)) = (a_|_ v (a ^ b))
10	2, 9	2or 67	. . . 4 (((a ->1 b) v b) v ((a ->1 b) v (a_|_ ^ b_|_))) = ((a_|_ v b) v (a_|_ v (a ^ b)))
11		id 58	. . . . . . 7 (a_|_ v b) = (a_|_ v b)
12	11	bile 134	. . . . . 6 (a_|_ v b) =< (a_|_ v b)
13		lear 153	. . . . . . 7 (a ^ b) =< b
14	13	lelor 158	. . . . . 6 (a_|_ v (a ^ b)) =< (a_|_ v b)
15	12, 14	lel2or 162	. . . . 5 ((a_|_ v b) v (a_|_ v (a ^ b))) =< (a_|_ v b)
16		leo 150	. . . . 5 (a_|_ v b) =< ((a_|_ v b) v (a_|_ v (a ^ b)))
17	15, 16	lebi 137	. . . 4 ((a_|_ v b) v (a_|_ v (a ^ b))) = (a_|_ v b)
18	10, 17	ax-r2 35	. . 3 (((a ->1 b) v b) v ((a ->1 b) v (a_|_ ^ b_|_))) = (a_|_ v b)
19	1, 18	ax-r2 35	. 2 ((a ->1 b) v (b v (a_|_ ^ b_|_))) = (a_|_ v b)
20		df-i2 44	. . 3 (a ->2 b) = (b v (a_|_ ^ b_|_))
21	20	lor 66	. 2 ((a ->1 b) v (a ->2 b)) = ((a ->1 b) v (b v (a_|_ ^ b_|_)))
22		df-i0 42	. 2 (a ->0 b) = (a_|_ v b)
23	19, 21, 22	3tr1 60	1 ((a ->1 b) v (a ->2 b)) = (a ->0 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->0 wi0 12   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  distoah2 921  distoah3 922  distoa 924  d3oa 975  oadist2b 988  oadist12 990

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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