Theorem cbvprodv 15867

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 15866	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1540  ∏cprod 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-seq 13974  df-prod 15857

This theorem is referenced by:  breprexp  34109  aks4d1  41421  mccl  44773  dvnprodlem3  45123  etransclem6  45415  etransclem37  45446  etransclem46  45455  ovnsubadd  45747  hoidmv1le  45769  hoidmvle  45775  hspmbl  45804  ovnovollem3  45833  vonn0ioo  45862  vonn0icc  45863