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Theorem oath1 1004
 Description: OA theorem.
Assertion
Ref Expression
oath1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))

Proof of Theorem oath1
StepHypRef Expression
1 oaliii 1001 . . . 4 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
21lan 77 . . 3 (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ∩ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ∩ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))
3 anidm 111 . . . 4 (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ∩ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
43ax-r1 35 . . 3 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ∩ ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))
5 anandir 115 . . 3 (((a2 b) ∩ (a2 c)) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ∩ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))))
62, 4, 53tr1 63 . 2 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ (a2 c)) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))
7 ax-a2 31 . . 3 ((bc) ∪ ((a2 b) ∩ (a2 c))) = (((a2 b) ∩ (a2 c)) ∪ (bc) )
87lan 77 . 2 (((a2 b) ∩ (a2 c)) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = (((a2 b) ∩ (a2 c)) ∩ (((a2 b) ∩ (a2 c)) ∪ (bc) ))
9 anabs 121 . 2 (((a2 b) ∩ (a2 c)) ∩ (((a2 b) ∩ (a2 c)) ∪ (bc) )) = ((a2 b) ∩ (a2 c))
106, 8, 93tr 65 1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  oalem2  1006  oadist2a  1007  oacom  1011  oacom3  1013  oagen1  1014
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