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Theorem axoa4 1034
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
StepHypRef Expression
1 u1lem9b 778 . . 3 a' =< (a ->1 d)
21leran 153 . 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 1033 . . . 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 59 . . . 4 (a ->1 d) = (a ->1 d)
5 id 59 . . . 4 (b ->1 d) = (b ->1 d)
6 id 59 . . . 4 (c ->1 d) = (c ->1 d)
73, 4, 5, 6oa4gto4u 976 . . 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
87oa4uto4 977 . 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
92, 8letr 137 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
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:  axoa4b  1035  axoa4d  1038
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