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 Description: Distributive inference derived from OA. (Contributed by NM, 17-Nov-1998.)
Hypothesis
Ref Expression
oadist2a.1 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oadist2a ((a2 b) ∩ (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))))

StepHypRef Expression
1 ax-a2 31 . . 3 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) = (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d)
21lan 77 . 2 ((a2 b) ∩ (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))) = ((a2 b) ∩ (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d))
3 ax-a2 31 . . . . . . 7 (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d) = (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c))))
4 oadist2a.1 . . . . . . 7 (d ∪ ((bc) →2 ((a2 b) ∩ (a2 c)))) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
53, 4bltr 138 . . . . . 6 (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d) ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
65lelan 167 . . . . 5 ((a2 b) ∩ (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d)) ≤ ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c))))
7 df-i0 43 . . . . . . . 8 ((bc) →0 ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
87lan 77 . . . . . . 7 ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
9 oath1 1004 . . . . . . 7 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
108, 9ax-r2 36 . . . . . 6 ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
11 leo 158 . . . . . . 7 ((a2 b) ∩ (a2 c)) ≤ (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
12 df-i2 45 . . . . . . . 8 ((bc) →2 ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) ))
1312ax-r1 35 . . . . . . 7 (((a2 b) ∩ (a2 c)) ∪ ((bc) ∩ ((a2 b) ∩ (a2 c)) )) = ((bc) →2 ((a2 b) ∩ (a2 c)))
1411, 13lbtr 139 . . . . . 6 ((a2 b) ∩ (a2 c)) ≤ ((bc) →2 ((a2 b) ∩ (a2 c)))
1510, 14bltr 138 . . . . 5 ((a2 b) ∩ ((bc) →0 ((a2 b) ∩ (a2 c)))) ≤ ((bc) →2 ((a2 b) ∩ (a2 c)))
166, 15letr 137 . . . 4 ((a2 b) ∩ (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d)) ≤ ((bc) →2 ((a2 b) ∩ (a2 c)))
1716distlem 188 . . 3 ((a2 b) ∩ (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d)) = (((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) ∪ ((a2 b) ∩ d))
18 ax-a2 31 . . 3 (((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) ∪ ((a2 b) ∩ d)) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))))
1917, 18ax-r2 36 . 2 ((a2 b) ∩ (((bc) →2 ((a2 b) ∩ (a2 c))) ∪ d)) = (((a2 b) ∩ d) ∪ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))))
202, 19ax-r2 36 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  ⊥ 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:  oadist2b  1008  oadist2  1009
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