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Theorem distlem 188
 Description: Distributive law inference (uses OL only).
Hypothesis
Ref Expression
distlem.1 (a ∩ (bc)) ≤ b
Assertion
Ref Expression
distlem (a ∩ (bc)) = ((ab) ∪ (ac))

Proof of Theorem distlem
StepHypRef Expression
1 lea 160 . . . 4 (a ∩ (bc)) ≤ a
2 distlem.1 . . . 4 (a ∩ (bc)) ≤ b
31, 2ler2an 173 . . 3 (a ∩ (bc)) ≤ (ab)
4 leo 158 . . 3 (ab) ≤ ((ab) ∪ (ac))
53, 4letr 137 . 2 (a ∩ (bc)) ≤ ((ab) ∪ (ac))
6 ledi 174 . 2 ((ab) ∪ (ac)) ≤ (a ∩ (bc))
75, 6lebi 145 1 (a ∩ (bc)) = ((ab) ∪ (ac))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by:  oadist2a  1007
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