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Theorem u12lem 771
 Description: Implication lemma. (Contributed by NM, 17-Nov-1998.)
Assertion
Ref Expression
u12lem ((a1 b) ∪ (a2 b)) = (a0 b)

Proof of Theorem u12lem
StepHypRef Expression
1 orordi 112 . . 3 ((a1 b) ∪ (b ∪ (ab ))) = (((a1 b) ∪ b) ∪ ((a1 b) ∪ (ab )))
2 u1lemob 630 . . . . 5 ((a1 b) ∪ b) = (ab)
3 df-i1 44 . . . . . . 7 (a1 b) = (a ∪ (ab))
43ax-r5 38 . . . . . 6 ((a1 b) ∪ (ab )) = ((a ∪ (ab)) ∪ (ab ))
5 or32 82 . . . . . . 7 ((a ∪ (ab)) ∪ (ab )) = ((a ∪ (ab )) ∪ (ab))
6 orabs 120 . . . . . . . 8 (a ∪ (ab )) = a
76ax-r5 38 . . . . . . 7 ((a ∪ (ab )) ∪ (ab)) = (a ∪ (ab))
85, 7ax-r2 36 . . . . . 6 ((a ∪ (ab)) ∪ (ab )) = (a ∪ (ab))
94, 8ax-r2 36 . . . . 5 ((a1 b) ∪ (ab )) = (a ∪ (ab))
102, 92or 72 . . . 4 (((a1 b) ∪ b) ∪ ((a1 b) ∪ (ab ))) = ((ab) ∪ (a ∪ (ab)))
11 id 59 . . . . . . 7 (ab) = (ab)
1211bile 142 . . . . . 6 (ab) ≤ (ab)
13 lear 161 . . . . . . 7 (ab) ≤ b
1413lelor 166 . . . . . 6 (a ∪ (ab)) ≤ (ab)
1512, 14lel2or 170 . . . . 5 ((ab) ∪ (a ∪ (ab))) ≤ (ab)
16 leo 158 . . . . 5 (ab) ≤ ((ab) ∪ (a ∪ (ab)))
1715, 16lebi 145 . . . 4 ((ab) ∪ (a ∪ (ab))) = (ab)
1810, 17ax-r2 36 . . 3 (((a1 b) ∪ b) ∪ ((a1 b) ∪ (ab ))) = (ab)
191, 18ax-r2 36 . 2 ((a1 b) ∪ (b ∪ (ab ))) = (ab)
20 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
2120lor 70 . 2 ((a1 b) ∪ (a2 b)) = ((a1 b) ∪ (b ∪ (ab )))
22 df-i0 43 . 2 (a0 b) = (ab)
2319, 21, 223tr1 63 1 ((a1 b) ∪ (a2 b)) = (a0 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →0 wi0 11   →1 wi1 12   →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 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:  distoah2  941  distoah3  942  distoa  944  d3oa  995  oadist2b  1008  oadist12  1010  lem4.6.6i1j2  1093
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