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Theorem cbvprodi 15867
Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
cbvprodi.1 𝑘𝐵
cbvprodi.2 𝑗𝐶
cbvprodi.3 (𝑗 = 𝑘𝐵 = 𝐶)
Assertion
Ref Expression
cbvprodi 𝑗𝐴 𝐵 = ∏𝑘𝐴 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 = 𝑘𝐵 = 𝐶)
2 nfcv 2897 . 2 𝑘𝐴
3 nfcv 2897 . 2 𝑗𝐴
4 cbvprodi.1 . 2 𝑘𝐵
5 cbvprodi.2 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 15865 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wnfc 2877  cprod 15855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-iota 6489  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seq 13973  df-prod 15856
This theorem is referenced by:  prodfc  15895  fprodcllemf  15908  prodsn  15912  prodsnf  15914  fprodm1s  15920  fprodp1s  15921  prodsns  15922  fprod2dlem  15930  fprodcom2  15934  fproddivf  15937  fprodsplitf  15938
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