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Theorem oi3oa3lem1 732
 Description: An attempt at the OA3 conjecture, which is true if (a ≡ b) = 1. (Contributed by Josiah Burroughs 27-May-04.)
Hypothesis
Ref Expression
oi3oa3lem1.1 1 = (ba)
Assertion
Ref Expression
oi3oa3lem1 (((a1 c) ∩ (b1 c)) ∪ (ab)) = 1

Proof of Theorem oi3oa3lem1
StepHypRef Expression
1 oi3oa3lem1.1 . . . . . 6 1 = (ba)
21r3a 440 . . . . 5 b = a
32ud1lem0b 256 . . . 4 (b1 c) = (a1 c)
43lan 77 . . 3 ((a1 c) ∩ (b1 c)) = ((a1 c) ∩ (a1 c))
52lan 77 . . 3 (ab) = (aa)
64, 52or 72 . 2 (((a1 c) ∩ (b1 c)) ∪ (ab)) = (((a1 c) ∩ (a1 c)) ∪ (aa))
7 anidm 111 . . 3 ((a1 c) ∩ (a1 c)) = (a1 c)
8 anidm 111 . . 3 (aa) = a
97, 82or 72 . 2 (((a1 c) ∩ (a1 c)) ∪ (aa)) = ((a1 c) ∪ a)
10 u1lemoa 620 . 2 ((a1 c) ∪ a) = 1
116, 9, 103tr 65 1 (((a1 c) ∩ (b1 c)) ∪ (ab)) = 1
 Colors of variables: term Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →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 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44 This theorem is referenced by:  oi3oa3  733
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