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Theorem axoa4 1034
 Description: The proper 4-variable OA law. (Contributed by NM, 20-Jul-1999.)
Assertion
Ref Expression
axoa4 (a ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ d

Proof of Theorem axoa4
StepHypRef Expression
1 u1lem9b 778 . . 3 a ≤ (a1 d)
21leran 153 . 2 (a ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ ((a1 d) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))))))
3 ax-4oa 1033 . . . 4 (((b1 d) →1 d) ∩ ((((b1 d) ∩ (a1 d)) ∪ (((b1 d) →1 d) ∩ ((a1 d) →1 d))) ∪ ((((b1 d) ∩ (c1 d)) ∪ (((b1 d) →1 d) ∩ ((c1 d) →1 d))) ∩ (((a1 d) ∩ (c1 d)) ∪ (((a1 d) →1 d) ∩ ((c1 d) →1 d)))))) ≤ ((a1 d) →1 d)
4 id 59 . . . 4 (a1 d) = (a1 d)
5 id 59 . . . 4 (b1 d) = (b1 d)
6 id 59 . . . 4 (c1 d) = (c1 d)
73, 4, 5, 6oa4gto4u 976 . . 3 ((a1 d) ∩ ((a1 d) ∪ ((b1 d) ∩ ((((a1 d) ∩ (b1 d)) ∪ ((a1 d) ∩ (b1 d))) ∪ ((((a1 d) ∩ (c1 d)) ∪ ((a1 d) ∩ (c1 d))) ∩ (((b1 d) ∩ (c1 d)) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ d
87oa4uto4 977 . 2 ((a1 d) ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ d
92, 8letr 137 1 (a ∩ (a ∪ (b ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))))) ≤ d
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ 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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