Theorem cbvprodv 15478

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 2904	. 2 ⊢ Ⅎ𝑘𝐴
3		nfcv 2904	. 2 ⊢ Ⅎ𝑗𝐴
4		nfcv 2904	. 2 ⊢ Ⅎ𝑘𝐵
5		nfcv 2904	. 2 ⊢ Ⅎ𝑗𝐶
6	1, 2, 3, 4, 5	cbvprod 15477	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

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