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Theorem sumeq2w 12478
 Description: Equality theorem for sum, when the class expressions and are equal everywhere. Proved using only Extensionality. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
sumeq2w
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem sumeq2w
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . . . 9
2 ifeq1 3735 . . . . . . . . . . 11
32alimi 1568 . . . . . . . . . 10
4 alral 2756 . . . . . . . . . 10
53, 4syl 16 . . . . . . . . 9
6 mpteq12 4280 . . . . . . . . 9
71, 5, 6sylancr 645 . . . . . . . 8
87seqeq3d 11323 . . . . . . 7
98breq1d 4214 . . . . . 6
109anbi2d 685 . . . . 5
1110rexbidv 2718 . . . 4
12 fvex 5734 . . . . . . . . . . . 12
13 nfcv 2571 . . . . . . . . . . . . 13
14 nfcsb1v 3275 . . . . . . . . . . . . . 14
15 nfcsb1v 3275 . . . . . . . . . . . . . 14
1614, 15nfeq 2578 . . . . . . . . . . . . 13
17 csbeq1a 3251 . . . . . . . . . . . . . 14
18 csbeq1a 3251 . . . . . . . . . . . . . 14
1917, 18eqeq12d 2449 . . . . . . . . . . . . 13
2013, 16, 19spcgf 3023 . . . . . . . . . . . 12
2112, 20ax-mp 8 . . . . . . . . . . 11
2221mpteq2dv 4288 . . . . . . . . . 10
2322seqeq3d 11323 . . . . . . . . 9
2423fveq1d 5722 . . . . . . . 8
2524eqeq2d 2446 . . . . . . 7
2625anbi2d 685 . . . . . 6
2726exbidv 1636 . . . . 5
2827rexbidv 2718 . . . 4
2911, 28orbi12d 691 . . 3
3029iotabidv 5431 . 2
31 df-sum 12472 . 2
32 df-sum 12472 . 2
3330, 31, 323eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948  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 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  ax-nul 4330 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-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-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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