Theorem oagen1b 995

Description: "Generalized" OA.

Hypotheses

Ref	Expression
oagen1b.1	d =< (a ->2 b)
oagen1b.2	e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oagen1b	(d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))

Proof of Theorem oagen1b

Step	Hyp	Ref	Expression
1		oagen1b.2	. . . 4 e =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2	1	oagen1 994	. . 3 ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
3	2	lan 70	. 2 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = (d ^ ((a ->2 b) ^ (a ->2 c)))
4		anass 69	. . . 4 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c)))))
5	4	ax-r1 34	. . 3 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c))))
6		oagen1b.1	. . . . 5 d =< (a ->2 b)
7	6	df2le2 128	. . . 4 (d ^ (a ->2 b)) = d
8	7	ran 71	. . 3 ((d ^ (a ->2 b)) ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
9	5, 8	ax-r2 35	. 2 (d ^ ((a ->2 b) ^ (e v ((a ->2 b) ^ (a ->2 c))))) = (d ^ (e v ((a ->2 b) ^ (a ->2 c))))
10		anass 69	. . . 4 ((d ^ (a ->2 b)) ^ (a ->2 c)) = (d ^ ((a ->2 b) ^ (a ->2 c)))
11	10	ax-r1 34	. . 3 (d ^ ((a ->2 b) ^ (a ->2 c))) = ((d ^ (a ->2 b)) ^ (a ->2 c))
12	7	ran 71	. . 3 ((d ^ (a ->2 b)) ^ (a ->2 c)) = (d ^ (a ->2 c))
13	11, 12	ax-r2 35	. 2 (d ^ ((a ->2 b) ^ (a ->2 c))) = (d ^ (a ->2 c))
14	3, 9, 13	3tr2 61	1 (d ^ (e v ((a ->2 b) ^ (a ->2 c)))) = (d ^ (a ->2 c))

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

This theorem is referenced by:  oadistd 1003

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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