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 Description: Distributive inference derived from OA.
Hypothesis
Ref Expression
oadist2b.1 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oadist2b ((a2 b) ∩ (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))))

Proof of Theorem oadist2b
StepHypRef Expression
1 oadist2b.1 . . . . 5 d ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
2 u12lem 771 . . . . . 6 (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((bc) →0 ((a2 b) ∩ (a2 c)))
32ax-r1 35 . . . . 5 ((bc) →0 ((a2 b) ∩ (a2 c))) = (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))
41, 3lbtr 139 . . . 4 d ≤ (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))
5 leor 159 . . . 4 ((bc) →2 ((a2 b) ∩ (a2 c))) ≤ (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))
64, 5lel2or 170 . . 3 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) ≤ (((bc) →1 ((a2 b) ∩ (a2 c))) ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))
76, 2lbtr 139 . 2 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
87oadist2a 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   ≤ wle 2   ∪ wo 6   ∩ wa 7   →0 wi0 11   →1 wi1 12   →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: (None)
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