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Theorem oa3-1to5 993
 Description: Derivation of an equivalent of the second "universal" 3-OA U2 from an equivalent of the first "universal" 3-OA U1. This shows that U2 is redundant in a system containg U1. The hypothesis is theorem oal1 1000. (Contributed by NM, 1-Jan-1999.)
Hypothesis
Ref Expression
oa3-1to5.1 ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
Assertion
Ref Expression
oa3-1to5 (c ∩ ((b1 c) ∪ ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ (b1 c)

Proof of Theorem oa3-1to5
StepHypRef Expression
1 leid 148 . . . . 5 (b1 c) ≤ (b1 c)
2 oa3-1to5.1 . . . . 5 ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))) ≤ (b1 c)
31, 2lel2or 170 . . . 4 ((b1 c) ∪ ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c))))) ≤ (b1 c)
43lelan 167 . . 3 (c ∩ ((b1 c) ∪ ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ (c ∩ (b1 c))
5 ax-a1 30 . . . . . . . 8 b = b
65ran 78 . . . . . . 7 (bc) = (b c)
76ax-r5 38 . . . . . 6 ((bc) ∪ (bc)) = ((b c) ∪ (bc))
8 ax-a2 31 . . . . . 6 ((b c) ∪ (bc)) = ((bc) ∪ (b c))
97, 8ax-r2 36 . . . . 5 ((bc) ∪ (bc)) = ((bc) ∪ (b c))
10 u1lemab 610 . . . . 5 ((b1 c) ∩ c) = ((bc) ∪ (bc))
11 u1lemab 610 . . . . 5 ((b1 c) ∩ c) = ((bc) ∪ (b c))
129, 10, 113tr1 63 . . . 4 ((b1 c) ∩ c) = ((b1 c) ∩ c)
13 ancom 74 . . . 4 (c ∩ (b1 c)) = ((b1 c) ∩ c)
14 ancom 74 . . . 4 (c ∩ (b1 c)) = ((b1 c) ∩ c)
1512, 13, 143tr1 63 . . 3 (c ∩ (b1 c)) = (c ∩ (b1 c))
164, 15lbtr 139 . 2 (c ∩ ((b1 c) ∪ ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ (c ∩ (b1 c))
17 lear 161 . 2 (c ∩ (b1 c)) ≤ (b1 c)
1816, 17letr 137 1 (c ∩ ((b1 c) ∪ ((a1 c) ∩ ((ab) ∪ ((a1 c) ∩ (b1 c)))))) ≤ (b1 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ 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 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: (None)
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