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Theorem oal2 999
 Description: Orthoarguesian law - →2 version.
Assertion
Ref Expression
oal2 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)

Proof of Theorem oal2
StepHypRef Expression
1 ax-3oa 998 . 2 ((b1 a ) ∩ ((bc ) ∪ ((b1 a ) ∩ (c1 a )))) ≤ (c1 a )
2 i2i1 267 . . 3 (a2 b) = (b1 a )
3 anor3 90 . . . . 5 (bc ) = (bc)
43ax-r1 35 . . . 4 (bc) = (bc )
5 i2i1 267 . . . . 5 (a2 c) = (c1 a )
62, 52an 79 . . . 4 ((a2 b) ∩ (a2 c)) = ((b1 a ) ∩ (c1 a ))
74, 62or 72 . . 3 ((bc) ∪ ((a2 b) ∩ (a2 c))) = ((bc ) ∪ ((b1 a ) ∩ (c1 a )))
82, 72an 79 . 2 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((b1 a ) ∩ ((bc ) ∪ ((b1 a ) ∩ (c1 a ))))
91, 8, 5le3tr1 140 1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998 This theorem depends on definitions:  df-a 40  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  oal1  1000  oaliii  1001  oagen2  1016  mloa  1018  oadistc0  1021
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