Theorem cbvprodi 12271

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 2386	. 2 ⊢ Ⅎ𝑘𝐴
3		nfcv 2386	. 2 ⊢ Ⅎ𝑗𝐴
4		cbvprodi.1	. 2 ⊢ Ⅎ𝑘𝐵
5		cbvprodi.2	. 2 ⊢ Ⅎ𝑗𝐶
6	1, 2, 3, 4, 5	cbvprod 12269	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1398  Ⅎwnfc 2373  ∏cprod 12261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-recs 6549  df-frec 6635  df-seqfrec 10834  df-proddc 12262

This theorem is referenced by:  prodfct  12298  prodsnf  12303  fprodm1s  12312  fprodp1s  12313  prodsns  12314  fprodcllemf  12324  fprod2dlemstep  12333  fprodcom2fi  12337  fproddivapf  12342  fprodsplitf  12343