Theorem cbvprodi 12066

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 2372	. 2 ⊢ Ⅎ𝑘𝐴
3		nfcv 2372	. 2 ⊢ Ⅎ𝑗𝐴
4		cbvprodi.1	. 2 ⊢ Ⅎ𝑘𝐵
5		cbvprodi.2	. 2 ⊢ Ⅎ𝑗𝐶
6	1, 2, 3, 4, 5	cbvprod 12064	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1395  Ⅎwnfc 2359  ∏cprod 12056

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-seqfrec 10665  df-proddc 12057

This theorem is referenced by:  prodfct  12093  prodsnf  12098  fprodm1s  12107  fprodp1s  12108  prodsns  12109  fprodcllemf  12119  fprod2dlemstep  12128  fprodcom2fi  12132  fproddivapf  12137  fprodsplitf  12138