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Theorem oa4to6dual 964
 Description: Lemma for orthoarguesian law (4-variable to 6-variable proof). (Contributed by NM, 19-Dec-1998.)
Hypotheses
Ref Expression
oa4to6lem.1 ab
oa4to6lem.2 cd
oa4to6lem.3 ef
oa4to6lem.4 g = (((ab) ∪ (cd)) ∪ (ef))
oa4to6lem.oa4 ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))) ≤ g
Assertion
Ref Expression
oa4to6dual (b ∩ (a ∪ (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))))))) ≤ g

Proof of Theorem oa4to6dual
StepHypRef Expression
1 oa4to6lem.1 . . 3 ab
2 oa4to6lem.2 . . 3 cd
3 oa4to6lem.3 . . 3 ef
4 oa4to6lem.4 . . 3 g = (((ab) ∪ (cd)) ∪ (ef))
51, 2, 3, 4oa4to6lem4 963 . 2 (b ∩ (a ∪ (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))))))) ≤ ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g))))))))
6 oa4to6lem.oa4 . 2 ((a1 g) ∩ (a ∪ (c ∩ (((ac) ∪ ((a1 g) ∩ (c1 g))) ∪ (((ae) ∪ ((a1 g) ∩ (e1 g))) ∩ ((ce) ∪ ((c1 g) ∩ (e1 g)))))))) ≤ g
75, 6letr 137 1 (b ∩ (a ∪ (c ∩ (((ac) ∪ (bd)) ∪ (((ae) ∪ (bf)) ∩ ((ce) ∪ (df))))))) ≤ g
 Colors of variables: term Syntax hints:   = wb 1   ≤ 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 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:  oa4to6  965  oa3-6to3  987  oa3-2to4  988  oa3-u1  991  oa3-u2  992
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