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Theorem oagen1b 1015
 Description: "Generalized" OA.
Hypotheses
Ref Expression
oagen1b.1 d ≤ (a2 b)
oagen1b.2 e ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oagen1b (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ (a2 c))

Proof of Theorem oagen1b
StepHypRef Expression
1 oagen1b.2 . . . 4 e ≤ ((bc) →0 ((a2 b) ∩ (a2 c)))
21oagen1 1014 . . 3 ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
32lan 77 . 2 (d ∩ ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c))))) = (d ∩ ((a2 b) ∩ (a2 c)))
4 anass 76 . . . 4 ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c)))))
54ax-r1 35 . . 3 (d ∩ ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c))))) = ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c))))
6 oagen1b.1 . . . . 5 d ≤ (a2 b)
76df2le2 136 . . . 4 (d ∩ (a2 b)) = d
87ran 78 . . 3 ((d ∩ (a2 b)) ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ (e ∪ ((a2 b) ∩ (a2 c))))
95, 8ax-r2 36 . 2 (d ∩ ((a2 b) ∩ (e ∪ ((a2 b) ∩ (a2 c))))) = (d ∩ (e ∪ ((a2 b) ∩ (a2 c))))
10 anass 76 . . . 4 ((d ∩ (a2 b)) ∩ (a2 c)) = (d ∩ ((a2 b) ∩ (a2 c)))
1110ax-r1 35 . . 3 (d ∩ ((a2 b) ∩ (a2 c))) = ((d ∩ (a2 b)) ∩ (a2 c))
127ran 78 . . 3 ((d ∩ (a2 b)) ∩ (a2 c)) = (d ∩ (a2 c))
1311, 12ax-r2 36 . 2 (d ∩ ((a2 b) ∩ (a2 c))) = (d ∩ (a2 c))
143, 9, 133tr2 64 1 (d ∩ (e ∪ ((a2 b) ∩ (a2 c)))) = (d ∩ (a2 c))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ 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:  oadistd  1023
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