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Theorem i3orlem2 553
 Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.)
Assertion
Ref Expression
i3orlem2 (ab) ≤ ((ac) →3 (bc))

Proof of Theorem i3orlem2
StepHypRef Expression
1 leo 158 . . 3 a ≤ (ac)
2 leo 158 . . 3 b ≤ (bc)
31, 2le2an 169 . 2 (ab) ≤ ((ac) ∩ (bc))
4 leor 159 . . . 4 ((ac) ∩ (bc)) ≤ (((ac) ∩ (ac) ) ∪ ((ac) ∩ (bc)))
5 ledi 174 . . . 4 (((ac) ∩ (ac) ) ∪ ((ac) ∩ (bc))) ≤ ((ac) ∩ ((ac) ∪ (bc)))
64, 5letr 137 . . 3 ((ac) ∩ (bc)) ≤ ((ac) ∩ ((ac) ∪ (bc)))
7 i3orlem1 552 . . 3 ((ac) ∩ ((ac) ∪ (bc))) ≤ ((ac) →3 (bc))
86, 7letr 137 . 2 ((ac) ∩ (bc)) ≤ ((ac) →3 (bc))
93, 8letr 137 1 (ab) ≤ ((ac) →3 (bc))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131 This theorem is referenced by:  i3orlem6  557
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