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Axiom ax-4oa 1033
Description: The proper 4-variable OA law.
Assertion
Ref Expression
ax-4oa ((a ->1 d) ^ (((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)

Detailed syntax breakdown of Axiom ax-4oa
StepHypRef Expression
1 wva . . . 4 term a
2 wvd . . . 4 term d
31, 2wi1 12 . . 3 term (a ->1 d)
4 wvb . . . . . 6 term b
51, 4wa 7 . . . . 5 term (a ^ b)
64, 2wi1 12 . . . . . 6 term (b ->1 d)
73, 6wa 7 . . . . 5 term ((a ->1 d) ^ (b ->1 d))
85, 7wo 6 . . . 4 term ((a ^ b) v ((a ->1 d) ^ (b ->1 d)))
9 wvc . . . . . . 7 term c
101, 9wa 7 . . . . . 6 term (a ^ c)
119, 2wi1 12 . . . . . . 7 term (c ->1 d)
123, 11wa 7 . . . . . 6 term ((a ->1 d) ^ (c ->1 d))
1310, 12wo 6 . . . . 5 term ((a ^ c) v ((a ->1 d) ^ (c ->1 d)))
144, 9wa 7 . . . . . 6 term (b ^ c)
156, 11wa 7 . . . . . 6 term ((b ->1 d) ^ (c ->1 d))
1614, 15wo 6 . . . . 5 term ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))
1713, 16wa 7 . . . 4 term (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d))))
188, 17wo 6 . . 3 term (((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)))))
193, 18wa 7 . 2 term ((a ->1 d) ^ (((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))))))
2019, 6wle 2 1 wff ((a ->1 d) ^ (((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
This axiom is referenced by:  axoa4  1034
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