Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvsum Structured version   Unicode version

Theorem cbvsum 12489
 Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
cbvsum.1
cbvsum.2
cbvsum.3
cbvsum.4
cbvsum.5
Assertion
Ref Expression
cbvsum
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem cbvsum
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvsum.2 . . . . . . . . . . . . 13
21nfcri 2566 . . . . . . . . . . . 12
3 cbvsum.4 . . . . . . . . . . . 12
4 nfcv 2572 . . . . . . . . . . . 12
52, 3, 4nfif 3763 . . . . . . . . . . 11
6 cbvsum.3 . . . . . . . . . . . . 13
76nfcri 2566 . . . . . . . . . . . 12
8 cbvsum.5 . . . . . . . . . . . 12
9 nfcv 2572 . . . . . . . . . . . 12
107, 8, 9nfif 3763 . . . . . . . . . . 11
11 eleq1 2496 . . . . . . . . . . . 12
12 cbvsum.1 . . . . . . . . . . . 12
13 eqidd 2437 . . . . . . . . . . . 12
1411, 12, 13ifbieq12d 3761 . . . . . . . . . . 11
155, 10, 14cbvmpt 4299 . . . . . . . . . 10
1615a1i 11 . . . . . . . . 9
1716seqeq3d 11331 . . . . . . . 8
1817breq1d 4222 . . . . . . 7
1918anbi2d 685 . . . . . 6
2019rexbidv 2726 . . . . 5
213, 8, 12cbvcsb 3255 . . . . . . . . . . . . 13
2221a1i 11 . . . . . . . . . . . 12
2322mpteq2dv 4296 . . . . . . . . . . 11
2423seqeq3d 11331 . . . . . . . . . 10
2524fveq1d 5730 . . . . . . . . 9
2625eqeq2d 2447 . . . . . . . 8
2726anbi2d 685 . . . . . . 7
2827exbidv 1636 . . . . . 6
2928rexbidv 2726 . . . . 5
3020, 29orbi12d 691 . . . 4
3130iotabidv 5439 . . 3
3231trud 1332 . 2
33 df-sum 12480 . 2
34 df-sum 12480 . 2
3532, 33, 343eqtr4i 2466 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 358   wa 359   wtru 1325  wex 1550   wceq 1652   wcel 1725  wnfc 2559  wrex 2706  csb 3251   wss 3320  cif 3739   class class class wbr 4212   cmpt 4266  cio 5416  wf1o 5453  cfv 5454  (class class class)co 6081  cc0 8990  c1 8991   caddc 8993  cn 10000  cz 10282  cuz 10488  cfz 11043   cseq 11323   cli 12278  csu 12479 This theorem is referenced by:  cbvsumv  12490  cbvsumi  12491  esumpfinvalf  24466 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 2417 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seq 11324  df-sum 12480
 Copyright terms: Public domain W3C validator