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Theorem sumeq1f 12474
 Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Hypotheses
Ref Expression
sumeq1f.1
sumeq1f.2
Assertion
Ref Expression
sumeq1f

Proof of Theorem sumeq1f
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3361 . . . . . 6
2 sumeq1f.1 . . . . . . . . . 10
3 sumeq1f.2 . . . . . . . . . 10
42, 3nfeq 2578 . . . . . . . . 9
5 simpl 444 . . . . . . . . . . 11
65eleq2d 2502 . . . . . . . . . 10
76ifbid 3749 . . . . . . . . 9
84, 7mpteq2da 4286 . . . . . . . 8
98seqeq3d 11323 . . . . . . 7
109breq1d 4214 . . . . . 6
111, 10anbi12d 692 . . . . 5
1211rexbidv 2718 . . . 4
13 f1oeq3 5659 . . . . . . 7
1413anbi1d 686 . . . . . 6
1514exbidv 1636 . . . . 5
1615rexbidv 2718 . . . 4
1712, 16orbi12d 691 . . 3
1817iotabidv 5431 . 2
19 df-sum 12472 . 2
20 df-sum 12472 . 2
2118, 19, 203eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359  wex 1550   wceq 1652   wcel 1725  wnfc 2558  wrex 2698  csb 3243   wss 3312  cif 3731   class class class wbr 4204   cmpt 4258  cio 5408  wf1o 5445  cfv 5446  (class class class)co 6073  cc0 8982  c1 8983   caddc 8985  cn 9992  cz 10274  cuz 10480  cfz 11035   cseq 11315   cli 12270  csu 12471 This theorem is referenced by:  sumeq1  12475 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11316  df-sum 12472
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