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Theorem cbvprodv 15478
Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
cbvprod.1 (𝑗 = 𝑘𝐵 = 𝐶)
Assertion
Ref Expression
cbvprodv 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗
Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvprodv
StepHypRef Expression
1 cbvprod.1 . 2 (𝑗 = 𝑘𝐵 = 𝐶)
2 nfcv 2904 . 2 𝑘𝐴
3 nfcv 2904 . 2 𝑗𝐴
4 nfcv 2904 . 2 𝑘𝐵
5 nfcv 2904 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 15477 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  cprod 15467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-iota 6338  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seq 13575  df-prod 15468
This theorem is referenced by:  breprexp  32325  mccl  42814  dvnprodlem3  43164  etransclem6  43456  etransclem37  43487  etransclem46  43496  ovnsubadd  43785  hoidmv1le  43807  hoidmvle  43813  hspmbl  43842  ovnovollem3  43871  vonn0ioo  43900  vonn0icc  43901
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