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Theorem 4oath1 1041
 Description: Proper 4-OA theorem. (Contributed by NM, 29-Dec-1998.)
Hypotheses
Ref Expression
4oa.1 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
4oa.2 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
Assertion
Ref Expression
4oath1 ((a1 d) ∩ f) = ((a1 d) ∩ (b1 d))

Proof of Theorem 4oath1
StepHypRef Expression
1 4oa.1 . . . . . 6 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
2 4oa.2 . . . . . 6 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
31, 24oaiii 1040 . . . . 5 ((a1 d) ∩ f) = ((b1 d) ∩ f)
43lan 77 . . . 4 (((a1 d) ∩ f) ∩ ((a1 d) ∩ f)) = (((a1 d) ∩ f) ∩ ((b1 d) ∩ f))
5 or32 82 . . . . . . 7 (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e) = (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))
62, 5ax-r2 36 . . . . . 6 f = (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))
76lan 77 . . . . 5 ((a1 d) ∩ f) = ((a1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))))
86lan 77 . . . . 5 ((b1 d) ∩ f) = ((b1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))))
97, 82an 79 . . . 4 (((a1 d) ∩ f) ∩ ((b1 d) ∩ f)) = (((a1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))) ∩ ((b1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))))
104, 9ax-r2 36 . . 3 (((a1 d) ∩ f) ∩ ((a1 d) ∩ f)) = (((a1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))) ∩ ((b1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))))
11 anidm 111 . . . 4 (((a1 d) ∩ f) ∩ ((a1 d) ∩ f)) = ((a1 d) ∩ f)
1211ax-r1 35 . . 3 ((a1 d) ∩ f) = (((a1 d) ∩ f) ∩ ((a1 d) ∩ f))
13 anandir 115 . . 3 (((a1 d) ∩ (b1 d)) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))) = (((a1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))) ∩ ((b1 d) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))))
1410, 12, 133tr1 63 . 2 ((a1 d) ∩ f) = (((a1 d) ∩ (b1 d)) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))))
15 ax-a2 31 . . 3 (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d))) = (((a1 d) ∩ (b1 d)) ∪ ((ab) ∪ e))
1615lan 77 . 2 (((a1 d) ∩ (b1 d)) ∩ (((ab) ∪ e) ∪ ((a1 d) ∩ (b1 d)))) = (((a1 d) ∩ (b1 d)) ∩ (((a1 d) ∩ (b1 d)) ∪ ((ab) ∪ e)))
17 anabs 121 . 2 (((a1 d) ∩ (b1 d)) ∩ (((a1 d) ∩ (b1 d)) ∪ ((ab) ∪ e))) = ((a1 d) ∩ (b1 d))
1814, 16, 173tr 65 1 ((a1 d) ∩ f) = ((a1 d) ∩ (b1 d))
 Colors of variables: term Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-4oa 1033 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  4oagen1  1042
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