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Theorem limeq 4160
 Description: Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
limeq (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))

Proof of Theorem limeq
StepHypRef Expression
1 ordeq 4155 . . 3 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
2 eleq2 2146 . . 3 (𝐴 = 𝐵 → (∅ ∈ 𝐴 ↔ ∅ ∈ 𝐵))
3 id 19 . . . 4 (𝐴 = 𝐵𝐴 = 𝐵)
4 unieq 3630 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
53, 4eqeq12d 2097 . . 3 (𝐴 = 𝐵 → (𝐴 = 𝐴𝐵 = 𝐵))
61, 2, 53anbi123d 1244 . 2 (𝐴 = 𝐵 → ((Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴) ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵𝐵 = 𝐵)))
7 dflim2 4153 . 2 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
8 dflim2 4153 . 2 (Lim 𝐵 ↔ (Ord 𝐵 ∧ ∅ ∈ 𝐵𝐵 = 𝐵))
96, 7, 83bitr4g 221 1 (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ∅c0 3267  ∪ cuni 3621  Ord word 4145  Lim wlim 4147 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-in 2988  df-ss 2995  df-uni 3622  df-tr 3896  df-iord 4149  df-ilim 4152 This theorem is referenced by:  limuni2  4180
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