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Theorem climeq 12354
 Description: Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
Hypotheses
Ref Expression
climeq.1
climeq.2
climeq.3
climeq.5
climeq.6
Assertion
Ref Expression
climeq
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem climeq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climeq.1 . . 3
2 climeq.5 . . 3
3 climeq.2 . . 3
4 climeq.6 . . 3
51, 2, 3, 4clim2 12291 . 2
6 climeq.3 . . 3
7 eqidd 2437 . . 3
81, 2, 6, 7clim2 12291 . 2
95, 8bitr4d 248 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2698  wrex 2699   class class class wbr 4205  cfv 5447  (class class class)co 6074  cc 8981   clt 9113   cmin 9284  cz 10275  cuz 10481  crp 10605  cabs 12032   cli 12271 This theorem is referenced by:  climmpt  12358  climres  12362  climshft  12363  climshft2  12369  isumclim3  12536  logtayl  20544  dfef2  20802  iprodclim3  25306  climexp  27699  stirlinglem14  27804 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-pre-lttri 9057  ax-pre-lttrn 9058 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-neg 9287  df-z 10276  df-uz 10482  df-clim 12275
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