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Theorem cbvprodi 15266
 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 2955 . 2 𝑘𝐴
3 nfcv 2955 . 2 𝑗𝐴
4 cbvprodi.1 . 2 𝑘𝐵
5 cbvprodi.2 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 15264 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2936  ∏cprod 15254 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-un 3886  df-in 3888  df-ss 3898  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-xp 5526  df-cnv 5528  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-iota 6284  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-seq 13368  df-prod 15255 This theorem is referenced by:  prodfc  15294  fprodcllemf  15307  prodsn  15311  prodsnf  15313  fprodm1s  15319  fprodp1s  15320  prodsns  15321  fprod2dlem  15329  fprodcom2  15333  fproddivf  15336  fprodsplitf  15337
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