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Theorem oadist2 1009
Description: Distributive inference derived from OA. (Contributed by NM, 17-Nov-1998.)
Hypothesis
Ref Expression
oadist2.1 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oadist2 ((a2 b) ∩ (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))))

Proof of Theorem oadist2
StepHypRef Expression
1 oadist2.1 . . 3 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((bc) →0 ((a2 b) ∩ (a2 c)))
21bile 142 . 2 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
32oadist2a 1007 1 ((a2 b) ∩ (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))))
Colors of variables: term
Syntax hints:   = wb 1  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:  oadist12  1010
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