Theorem oa6 1015

Description: Derivation of 6-variable orthoarguesian law from 4-variable version.

Hypotheses

Ref	Expression
oa6.1	a =< b_|_
oa6.2	c =< d_|_
oa6.3	e =< f_|_

Assertion

Ref	Expression
oa6	(((a v b) ^ (c v d)) ^ (e v f)) =< (b v (a ^ (c v (((a v c) ^ (b v d)) ^ (((a v e) ^ (b v f)) v ((c v e) ^ (d v f)))))))

Proof of Theorem oa6

Step	Hyp	Ref	Expression
1		oa6.1	. 2 a =< b_|_
2		oa6.2	. 2 c =< d_|_
3		oa6.3	. 2 e =< f_|_
4		id 58	. 2 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_)) = (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))
5		id 58	. 2 a_|_ = a_|_
6		id 58	. 2 c_|_ = c_|_
7		id 58	. 2 e_|_ = e_|_
8		axoa4b 1014	. 2 ((a_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))) ^ (a_|_ v (c_|_ ^ (((a_|_ ^ c_|_) v ((a_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))) ^ (c_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))))) v (((a_|_ ^ e_|_) v ((a_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))) ^ (e_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))))) ^ ((c_|_ ^ e_|_) v ((c_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))) ^ (e_|_ ->1 (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_)))))))))) =< (((a_|_ ^ b_|_) v (c_|_ ^ d_|_)) v (e_|_ ^ f_|_))
9	1, 2, 3, 4, 5, 6, 7, 8	oa4to6 945	1 (((a v b) ^ (c v d)) ^ (e v f)) =< (b v (a ^ (c v (((a v c) ^ (b v d)) ^ (((a v e) ^ (b v f)) v ((c v e) ^ (d v f)))))))

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7

This theorem is referenced by:  axoa4a 1016

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421  ax-4oa 1012

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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