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Theorem cbvprodv 15864
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 2901 . 2 𝑘𝐴
3 nfcv 2901 . 2 𝑗𝐴
4 nfcv 2901 . 2 𝑘𝐵
5 nfcv 2901 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 15863 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cprod 15853
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5681  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seq 13971  df-prod 15854
This theorem is referenced by:  breprexp  33943  aks4d1  41260  mccl  44612  dvnprodlem3  44962  etransclem6  45254  etransclem37  45285  etransclem46  45294  ovnsubadd  45586  hoidmv1le  45608  hoidmvle  45614  hspmbl  45643  ovnovollem3  45672  vonn0ioo  45701  vonn0icc  45702
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