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Theorem axoa4d 1038
 Description: Proper 4-variable OA law variant. (Contributed by NM, 24-Dec-1998.)
Assertion
Ref Expression
axoa4d (a ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))) ≤ (b1 d)

Proof of Theorem axoa4d
StepHypRef Expression
1 oa4dcom 970 . . 3 (a ∩ (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))))) = (a ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))))
21ax-r1 35 . 2 (a ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))) = (a ∩ (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d))))))
3 axoa4 1034 . . 3 (b ∩ (b ∪ (a ∩ (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))))))) ≤ d
43oa4ctod 968 . 2 (a ∩ (((ba) ∪ ((b1 d) ∩ (a1 d))) ∪ (((bc) ∪ ((b1 d) ∩ (c1 d))) ∩ ((ac) ∪ ((a1 d) ∩ (c1 d)))))) ≤ (b1 d)
52, 4bltr 138 1 (a ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d)))))) ≤ (b1 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: (None)
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