Theorem cbvprodi 11501

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

Hypotheses

Ref	Expression
cbvprodi.1	|- F/_ k B
cbvprodi.2	|- F/_ j C
cbvprodi.3	|- ( j = k -> B = C )

Assertion

Ref	Expression
cbvprodi	|- prod_ j e. A B = prod_ k e. 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 2308	. 2 |- F/_ k A
3		nfcv 2308	. 2 |- F/_ j A
4		cbvprodi.1	. 2 |- F/_ k B
5		cbvprodi.2	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvprod 11499	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1343   F/_wnfc 2295   prod_cprod 11491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359  df-seqfrec 10381  df-proddc 11492

This theorem is referenced by:  prodfct  11528  prodsnf  11533  fprodm1s  11542  fprodp1s  11543  prodsns  11544  fprodcllemf  11554  fprod2dlemstep  11563  fprodcom2fi  11567  fproddivapf  11572  fprodsplitf  11573