Theorem oadist 999

Description: Distributive law derived from OAL.

Hypothesis

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

Assertion

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

Proof of Theorem oadist

Step	Hyp	Ref	Expression
1		oadist.1	. . . . 5 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2	1	oagen1 994	. . . 4 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))
3	2	bile 134	. . 3 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (a ->2 c))
4		anidm 103	. . . . . . 7 ((a ->2 b) ^ (a ->2 b)) = (a ->2 b)
5	4	ax-r1 34	. . . . . 6 (a ->2 b) = ((a ->2 b) ^ (a ->2 b))
6	5	ran 71	. . . . 5 ((a ->2 b) ^ (a ->2 c)) = (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c))
7		anass 69	. . . . 5 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
8	6, 7	ax-r2 35	. . . 4 ((a ->2 b) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
9		leor 151	. . . 4 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) =< (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
10	8, 9	bltr 130	. . 3 ((a ->2 b) ^ (a ->2 c)) =< (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
11	3, 10	letr 129	. 2 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) =< (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))
12		ledi 166	. 2 (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c))))
13	11, 12	lebi 137	1 ((a ->2 b) ^ (d v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))))

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

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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