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Theorem cbvprodi 15951
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 2905 . 2 𝑘𝐴
3 nfcv 2905 . 2 𝑗𝐴
4 cbvprodi.1 . 2 𝑘𝐵
5 cbvprodi.2 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 15949 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wnfc 2890  cprod 15939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043  df-prod 15940
This theorem is referenced by:  prodfc  15981  fprodcllemf  15994  prodsn  15998  prodsnf  16000  fprodm1s  16006  fprodp1s  16007  prodsns  16008  fprod2dlem  16016  fprodcom2  16020  fproddivf  16023  fprodsplitf  16024
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