Theorem oal1 980

Description: Orthoarguesian law - ->1 version derived from ->1 version.

Assertion

Ref	Expression
oal1	((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)

Proof of Theorem oal1

Step	Hyp	Ref	Expression
1		oal2 979	. 2 ((c_|_ ->2 a_|_) ^ ((a_|_ v b_|_)_|_ v ((c_|_ ->2 a_|_) ^ (c_|_ ->2 b_|_)))) =< (c_|_ ->2 b_|_)
2		i1i2 258	. . 3 (a ->1 c) = (c_|_ ->2 a_|_)
3		df-a 39	. . . 4 (a ^ b) = (a_|_ v b_|_)_|_
4		i1i2 258	. . . . 5 (b ->1 c) = (c_|_ ->2 b_|_)
5	2, 4	2an 72	. . . 4 ((a ->1 c) ^ (b ->1 c)) = ((c_|_ ->2 a_|_) ^ (c_|_ ->2 b_|_))
6	3, 5	2or 67	. . 3 ((a ^ b) v ((a ->1 c) ^ (b ->1 c))) = ((a_|_ v b_|_)_|_ v ((c_|_ ->2 a_|_) ^ (c_|_ ->2 b_|_)))
7	2, 6	2an 72	. 2 ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) = ((c_|_ ->2 a_|_) ^ ((a_|_ v b_|_)_|_ v ((c_|_ ->2 a_|_) ^ (c_|_ ->2 b_|_))))
8	1, 7, 4	le3tr1 132	1 ((a ->1 c) ^ ((a ^ b) v ((a ->1 c) ^ (b ->1 c)))) =< (b ->1 c)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  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-i1 43  df-i2 44  df-le1 122  df-le2 123

metamath.org