Theorem oagen1 994

Description: "Generalized" OA.

Hypothesis

Ref	Expression
oagen1.1	d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oagen1	((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))

Proof of Theorem oagen1

Step	Hyp	Ref	Expression
1		oagen1.1	. . . . . . 7 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2		df-i0 42	. . . . . . 7 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
3	1, 2	lbtr 131	. . . . . 6 d =< ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
4	3	leror 144	. . . . 5 (d v ((a ->2 b) ^ (a ->2 c))) =< (((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ (a ->2 c)))
5		ax-a3 31	. . . . . 6 (((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ (a ->2 c))))
6		oridm 102	. . . . . . 7 (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ (a ->2 c))
7	6	lor 66	. . . . . 6 ((b v c)_|_ v (((a ->2 b) ^ (a ->2 c)) v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
8	5, 7	ax-r2 35	. . . . 5 (((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
9	4, 8	lbtr 131	. . . 4 (d v ((a ->2 b) ^ (a ->2 c))) =< ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
10	9	lelan 159	. . 3 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
11		oath1 984	. . 3 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
12	10, 11	lbtr 131	. 2 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (a ->2 c))
13		lea 152	. . 3 ((a ->2 b) ^ (a ->2 c)) =< (a ->2 b)
14		leor 151	. . 3 ((a ->2 b) ^ (a ->2 c)) =< (d v ((a ->2 b) ^ (a ->2 c)))
15	13, 14	ler2an 165	. 2 ((a ->2 b) ^ (a ->2 c)) =< ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c))))
16	12, 15	lebi 137	1 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))

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

This theorem is referenced by:  oagen1b 995  oadist 999  oadistb 1000

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-3oa 978

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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