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Theorem cbvprodv 14485
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 2751 . 2 𝑘𝐴
3 nfcv 2751 . 2 𝑗𝐴
4 nfcv 2751 . 2 𝑘𝐵
5 nfcv 2751 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 14484 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  cprod 14474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-prod 14475
This theorem is referenced by:  mccl  38665  dvnprodlem3  38838  etransclem6  39133  etransclem37  39164  etransclem46  39173  ovnsubadd  39462  hoidmv1le  39484  hoidmvle  39490  hspmbl  39519  ovnovollem3  39548  vonn0ioo  39578  vonn0icc  39579
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