Theorem oadist2b 988

Description: Distributive inference derived from OA.

Hypothesis

Ref	Expression
oadist2b.1	d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))

Assertion

Ref	Expression
oadist2b	((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))

Proof of Theorem oadist2b

Step	Hyp	Ref	Expression
1		oadist2b.1	. . . . 5 d =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
2		u12lem 753	. . . . . 6 (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
3	2	ax-r1 34	. . . . 5 ((b v c) ->0 ((a ->2 b) ^ (a ->2 c))) = (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
4	1, 3	lbtr 131	. . . 4 d =< (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
5		leor 151	. . . 4 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) =< (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
6	4, 5	lel2or 162	. . 3 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< (((b v c) ->1 ((a ->2 b) ^ (a ->2 c))) v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))
7	6, 2	lbtr 131	. 2 (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) =< ((b v c) ->0 ((a ->2 b) ^ (a ->2 c)))
8	7	oadist2a 987	1 ((a ->2 b) ^ (d v ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ d) v ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))))

Syntax hints:   = wb 1   =< wle 2   v wo 6   ^ wa 7   ->0 wi0 12   ->1 wi1 13   ->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

metamath.org