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Theorem limeq 4059
 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 4054 . . 3 (A = B → (Ord A ↔ Ord B))
2 eleq2 2079 . . 3 (A = B → (∅ A ↔ ∅ B))
3 id 19 . . . 4 (A = BA = B)
4 unieq 3559 . . . 4 (A = B A = B)
53, 4eqeq12d 2032 . . 3 (A = B → (A = AB = B))
61, 2, 53anbi123d 1190 . 2 (A = B → ((Ord A A A = A) ↔ (Ord B B B = B)))
7 dflim2 4052 . 2 (Lim A ↔ (Ord A A A = A))
8 dflim2 4052 . 2 (Lim B ↔ (Ord B B B = B))
96, 7, 83bitr4g 212 1 (A = B → (Lim A ↔ Lim B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 871   = wceq 1226   ∈ wcel 1370  ∅c0 3197  ∪ cuni 3550  Ord word 4044  Lim wlim 4046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-in 2897  df-ss 2904  df-uni 3551  df-tr 3825  df-iord 4048  df-ilim 4051 This theorem is referenced by:  limuni2  4079
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