Theorem oa4ctod 948

Description: Derivation of 4-OA law variant.

Hypothesis

Ref	Expression
oa4ctod.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

Assertion

Ref	Expression
oa4ctod	(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)

Proof of Theorem oa4ctod

Step	Hyp	Ref	Expression
1		oa4ctod.1	. 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)))))))) =< d
2	1	oat 907	1 (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)

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

This theorem is referenced by:  axoa4d 1017

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-r3 421

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

metamath.org