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Theorem oaidlem1 294
 Description: Lemma for OA identity-like law. (Contributed by NM, 22-Jan-1999.)
Hypothesis
Ref Expression
oaidlem1.1 (ab) ≤ c
Assertion
Ref Expression
oaidlem1 (a ∪ (b1 c)) = 1

Proof of Theorem oaidlem1
StepHypRef Expression
1 df-i1 44 . . 3 (b1 c) = (b ∪ (bc))
21lor 70 . 2 (a ∪ (b1 c)) = (a ∪ (b ∪ (bc)))
3 oran3 93 . . . 4 (ab ) = (ab)
43ax-r5 38 . . 3 ((ab ) ∪ (bc)) = ((ab) ∪ (bc))
5 ax-a3 32 . . 3 ((ab ) ∪ (bc)) = (a ∪ (b ∪ (bc)))
6 lear 161 . . . . 5 (ab) ≤ b
7 oaidlem1.1 . . . . 5 (ab) ≤ c
86, 7ler2an 173 . . . 4 (ab) ≤ (bc)
98sklem 230 . . 3 ((ab) ∪ (bc)) = 1
104, 5, 93tr2 64 . 2 (a ∪ (b ∪ (bc))) = 1
112, 10ax-r2 36 1 (a ∪ (b1 c)) = 1
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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