Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvprodi Structured version   Unicode version

Theorem cbvprodi 29024
Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
cbvprodi.1  |-  F/_ k B
cbvprodi.2  |-  F/_ j C
cbvprodi.3  |-  ( j  =  k  ->  B  =  C )
Assertion
Ref Expression
cbvprodi  |-  prod_ j  e.  A  B  =  prod_ k  e.  A  C
Distinct variable group:    j, k, A
Allowed substitution hints:    B( j, k)    C( j, k)

Proof of Theorem cbvprodi
StepHypRef Expression
1 cbvprodi.3 . 2  |-  ( j  =  k  ->  B  =  C )
2 nfcv 2605 . 2  |-  F/_ k A
3 nfcv 2605 . 2  |-  F/_ j A
4 cbvprodi.1 . 2  |-  F/_ k B
5 cbvprodi.2 . 2  |-  F/_ j C
61, 2, 3, 4, 5cbvprod 29022 1  |-  prod_ j  e.  A  B  =  prod_ k  e.  A  C
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383   F/_wnfc 2591   prod_cprod 29012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-recs 7044  df-rdg 7078  df-seq 12087  df-prod 29013
This theorem is referenced by:  prodfc  29052  prodsn  29067  fprodm1s  29074  fprodp1s  29075  prodsns  29076  fprod2dlem  29085  fprodcom2  29089
  Copyright terms: Public domain W3C validator