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Theorem axoa4d 1038
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
StepHypRef Expression
1 oa4dcom 970 . . 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))))))
21ax-r1 35 . 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 1034 . . 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
43oa4ctod 968 . 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)
52, 4bltr 138 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)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-4oa 1033
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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