Theorem axoa4d 1017

Description: Proper 4-variable OA law variant.

Assertion

Ref	Expression
axoa4d	(a ^ (((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)))))) =< (b_|_ ->1 d)

Proof of Theorem axoa4d

Step	Hyp	Ref	Expression
1		oa4dcom 950	. . 3 (a ^ (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))))) = (a ^ (((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))))))
2	1	ax-r1 34	. 2 (a ^ (((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 ^ (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d))))))
3		axoa4 1013	. . 3 (b_|_ ^ (b v (a ^ (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))))))) =< d
4	3	oa4ctod 948	. 2 (a ^ (((b ^ a) v ((b ->1 d) ^ (a ->1 d))) v (((b ^ c) v ((b ->1 d) ^ (c ->1 d))) ^ ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))))) =< (b_|_ ->1 d)
5	2, 4	bltr 130	1 (a ^ (((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)))))) =< (b_|_ ->1 d)

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

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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