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Theorem oa3to4lem5 949
 Description: Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable proof).
Hypothesis
Ref Expression
oa3to4lem5.1 ((ab) ∩ (cd)) ≤ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))
Assertion
Ref Expression
oa3to4lem5 ((ba) ∩ (dc)) ≤ (a ∪ (b ∩ (d ∪ ((bd) ∩ (ac)))))

Proof of Theorem oa3to4lem5
StepHypRef Expression
1 oa3to4lem5.1 . 2 ((ab) ∩ (cd)) ≤ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))
2 ax-a2 31 . . 3 (ba) = (ab)
3 ax-a2 31 . . 3 (dc) = (cd)
42, 32an 79 . 2 ((ba) ∩ (dc)) = ((ab) ∩ (cd))
5 ancom 74 . . . . 5 ((bd) ∩ (ac)) = ((ac) ∩ (bd))
65lor 70 . . . 4 (d ∪ ((bd) ∩ (ac))) = (d ∪ ((ac) ∩ (bd)))
76lan 77 . . 3 (b ∩ (d ∪ ((bd) ∩ (ac)))) = (b ∩ (d ∪ ((ac) ∩ (bd))))
87lor 70 . 2 (a ∪ (b ∩ (d ∪ ((bd) ∩ (ac))))) = (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))
91, 4, 8le3tr1 140 1 ((ba) ∩ (dc)) ≤ (a ∪ (b ∩ (d ∪ ((bd) ∩ (ac)))))
 Colors of variables: term Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ wa 7 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 This theorem depends on definitions:  df-a 40  df-le1 130  df-le2 131 This theorem is referenced by:  oa3to4  951
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