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Theorem oagen2 1016
 Description: "Generalized" OA.
Hypothesis
Ref Expression
oagen2.1 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oagen2 ((a2 b) ∩ d) ≤ (a2 c)

Proof of Theorem oagen2
StepHypRef Expression
1 oagen2.1 . . . 4 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
2 df-i0 43 . . . 4 ((bc) →0 ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
31, 2lbtr 139 . . 3 d ≤ ((bc) ∪ ((a2 b) ∩ (a2 c)))
43lelan 167 . 2 ((a2 b) ∩ d) ≤ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
5 oal2 999 . 2 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
64, 5letr 137 1 ((a2 b) ∩ d) ≤ (a2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →2 wi2 13 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  ax-3oa 998 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  oagen2b  1017
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