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Theorem oacom3 1013
 Description: Commutation law requiring OA.
Hypotheses
Ref Expression
oacom3.1 (d ∩ (a2 b)) C ((bc) →0 ((a2 b) ∩ (a2 c)))
oacom3.2 d C (a2 b)
Assertion
Ref Expression
oacom3 d C ((a2 b) ∩ (a2 c))

Proof of Theorem oacom3
StepHypRef Expression
1 oacom3.2 . . . . 5 d C (a2 b)
21comcom 453 . . . 4 (a2 b) C d
3 ancom 74 . . . . . 6 ((a2 b) ∩ d) = (d ∩ (a2 b))
4 oacom3.1 . . . . . 6 (d ∩ (a2 b)) C ((bc) →0 ((a2 b) ∩ (a2 c)))
53, 4bctr 181 . . . . 5 ((a2 b) ∩ d) C ((bc) →0 ((a2 b) ∩ (a2 c)))
65comcom 453 . . . 4 ((bc) →0 ((a2 b) ∩ (a2 c))) C ((a2 b) ∩ d)
72, 6gsth2 490 . . 3 (((bc) →0 ((a2 b) ∩ (a2 c))) ∩ (a2 b)) C d
87comcom 453 . 2 d C (((bc) →0 ((a2 b) ∩ (a2 c))) ∩ (a2 b))
9 df-i0 43 . . . 4 ((bc) →0 ((a2 b) ∩ (a2 c))) = ((bc) ∪ ((a2 b) ∩ (a2 c)))
109ran 78 . . 3 (((bc) →0 ((a2 b) ∩ (a2 c))) ∩ (a2 b)) = (((bc) ∪ ((a2 b) ∩ (a2 c))) ∩ (a2 b))
11 ancom 74 . . 3 (((bc) ∪ ((a2 b) ∩ (a2 c))) ∩ (a2 b)) = ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
12 oath1 1004 . . 3 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
1310, 11, 123tr 65 . 2 (((bc) →0 ((a2 b) ∩ (a2 c))) ∩ (a2 b)) = ((a2 b) ∩ (a2 c))
148, 13cbtr 182 1 d C ((a2 b) ∩ (a2 c))
 Colors of variables: term Syntax hints:   C wc 3  ⊥ 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-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-3oa 998 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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