Theorem cbvprodv 15810

Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypothesis

Ref	Expression
cbvprod.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvprodv	⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑘   𝐶,𝑗

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑘)

Proof of Theorem cbvprodv

Step	Hyp	Ref	Expression
1		cbvprod.1	. 2 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
2		nfcv 2902	. 2 ⊢ Ⅎ𝑘𝐴
3		nfcv 2902	. 2 ⊢ Ⅎ𝑗𝐴
4		nfcv 2902	. 2 ⊢ Ⅎ𝑘𝐵
5		nfcv 2902	. 2 ⊢ Ⅎ𝑗𝐶
6	1, 2, 3, 4, 5	cbvprod 15809	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1541  ∏cprod 15799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-xp 5644  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-iota 6453  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-seq 13917  df-prod 15800

This theorem is referenced by:  breprexp  33335  aks4d1  40619  mccl  43959  dvnprodlem3  44309  etransclem6  44601  etransclem37  44632  etransclem46  44641  ovnsubadd  44933  hoidmv1le  44955  hoidmvle  44961  hspmbl  44990  ovnovollem3  45019  vonn0ioo  45048  vonn0icc  45049