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Theorem cbvprodv 14627
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 2762 . 2 𝑘𝐴
3 nfcv 2762 . 2 𝑗𝐴
4 nfcv 2762 . 2 𝑘𝐵
5 nfcv 2762 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 14626 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  cprod 14616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-iota 5839  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seq 12785  df-prod 14617
This theorem is referenced by:  breprexp  30685  mccl  39630  dvnprodlem3  39926  etransclem6  40220  etransclem37  40251  etransclem46  40260  ovnsubadd  40549  hoidmv1le  40571  hoidmvle  40577  hspmbl  40606  ovnovollem3  40635  vonn0ioo  40664  vonn0icc  40665
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