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Theorem i3orlem5 556
 Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.)
Assertion
Ref Expression
i3orlem5 ((ab ) ∩ c ) ≤ ((ac) →3 (bc))

Proof of Theorem i3orlem5
StepHypRef Expression
1 leo 158 . 2 ((ac) ∩ (bc) ) ≤ (((ac) ∩ (bc) ) ∪ (((ac) ∪ (bc)) ∩ ((ac) ∪ ((ac) ∩ (bc)))))
2 anandir 115 . . 3 ((ab ) ∩ c ) = ((ac ) ∩ (bc ))
3 oran 87 . . . . . 6 (ac) = (ac )
43con2 67 . . . . 5 (ac) = (ac )
54ax-r1 35 . . . 4 (ac ) = (ac)
6 oran 87 . . . . . 6 (bc) = (bc )
76con2 67 . . . . 5 (bc) = (bc )
87ax-r1 35 . . . 4 (bc ) = (bc)
95, 82an 79 . . 3 ((ac ) ∩ (bc )) = ((ac) ∩ (bc) )
102, 9ax-r2 36 . 2 ((ab ) ∩ c ) = ((ac) ∩ (bc) )
11 df2i3 498 . 2 ((ac) →3 (bc)) = (((ac) ∩ (bc) ) ∪ (((ac) ∪ (bc)) ∩ ((ac) ∪ ((ac) ∩ (bc)))))
121, 10, 11le3tr1 140 1 ((ab ) ∩ c ) ≤ ((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-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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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