Theorem cbvprodi 15901

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

Hypotheses

Ref	Expression
cbvprodi.1	⊢ Ⅎ𝑘𝐵
cbvprodi.2	⊢ Ⅎ𝑗𝐶
cbvprodi.3	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)

Assertion

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

Distinct variable group:   𝑗,𝑘,𝐴

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

Proof of Theorem cbvprodi

Step	Hyp	Ref	Expression
1		cbvprodi.3	. 2 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
2		nfcv 2899	. 2 ⊢ Ⅎ𝑘𝐴
3		nfcv 2899	. 2 ⊢ Ⅎ𝑗𝐴
4		cbvprodi.1	. 2 ⊢ Ⅎ𝑘𝐵
5		cbvprodi.2	. 2 ⊢ Ⅎ𝑗𝐶
6	1, 2, 3, 4, 5	cbvprod 15899	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1533  Ⅎwnfc 2879  ∏cprod 15889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-iota 6505  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-seq 14007  df-prod 15890

This theorem is referenced by:  prodfc  15929  fprodcllemf  15942  prodsn  15946  prodsnf  15948  fprodm1s  15954  fprodp1s  15955  prodsns  15956  fprod2dlem  15964  fprodcom2  15968  fproddivf  15971  fprodsplitf  15972