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Theorem oa6 1036
 Description: Derivation of 6-variable orthoarguesian law from 4-variable version.
Hypotheses
Ref Expression
oa6.1 ab
oa6.2 cd
oa6.3 ef
Assertion
Ref Expression
oa6 (((ab) ∩ (cd)) ∩ (ef)) ≤ (b ∪ (a ∩ (c ∪ (((ac) ∩ (bd)) ∩ (((ae) ∩ (bf)) ∪ ((ce) ∩ (df)))))))

Proof of Theorem oa6
StepHypRef Expression
1 oa6.1 . 2 ab
2 oa6.2 . 2 cd
3 oa6.3 . 2 ef
4 id 59 . 2 (((ab ) ∪ (cd )) ∪ (ef )) = (((ab ) ∪ (cd )) ∪ (ef ))
5 id 59 . 2 a = a
6 id 59 . 2 c = c
7 id 59 . 2 e = e
8 axoa4b 1035 . 2 ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (a ∪ (c ∩ (((ac ) ∪ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (c1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∪ (((ae ) ∪ ((a1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef ))))) ∩ ((ce ) ∪ ((c1 (((ab ) ∪ (cd )) ∪ (ef ))) ∩ (e1 (((ab ) ∪ (cd )) ∪ (ef )))))))))) ≤ (((ab ) ∪ (cd )) ∪ (ef ))
91, 2, 3, 4, 5, 6, 7, 8oa4to6 965 1 (((ab) ∩ (cd)) ∩ (ef)) ≤ (b ∪ (a ∩ (c ∪ (((ac) ∩ (bd)) ∩ (((ae) ∩ (bf)) ∪ ((ce) ∩ (df)))))))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7 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:  axoa4a  1037
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