Theorem axoa4 1013

Description: The proper 4-variable OA law.

Assertion

Ref	Expression
axoa4	(a_|_ ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< d

Proof of Theorem axoa4

Step	Hyp	Ref	Expression
1		u1lem9b 760	. . 3 a_|_ =< (a ->1 d)
2	1	leran 145	. 2 (a_|_ ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< ((a ->1 d) ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))))))
3		ax-4oa 1012	. . . 4 (((b ->1 d) ->1 d) ^ ((((b ->1 d) ^ (a ->1 d)) v (((b ->1 d) ->1 d) ^ ((a ->1 d) ->1 d))) v ((((b ->1 d) ^ (c ->1 d)) v (((b ->1 d) ->1 d) ^ ((c ->1 d) ->1 d))) ^ (((a ->1 d) ^ (c ->1 d)) v (((a ->1 d) ->1 d) ^ ((c ->1 d) ->1 d)))))) =< ((a ->1 d) ->1 d)
4		id 58	. . . 4 (a ->1 d) = (a ->1 d)
5		id 58	. . . 4 (b ->1 d) = (b ->1 d)
6		id 58	. . . 4 (c ->1 d) = (c ->1 d)
7	3, 4, 5, 6	oa4gto4u 956	. . 3 ((a ->1 d) ^ ((a_|_ ->1 d) v ((b_|_ ->1 d) ^ ((((a ->1 d) ^ (b ->1 d)) v ((a_|_ ->1 d) ^ (b_|_ ->1 d))) v ((((a ->1 d) ^ (c ->1 d)) v ((a_|_ ->1 d) ^ (c_|_ ->1 d))) ^ (((b ->1 d) ^ (c ->1 d)) v ((b_|_ ->1 d) ^ (c_|_ ->1 d)))))))) =< d
8	7	oa4uto4 957	. 2 ((a ->1 d) ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< d
9	2, 8	letr 129	1 (a_|_ ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< d

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

This theorem is referenced by:  axoa4b 1014  axoa4d 1017

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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