Theorem cbvprodv 11705

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	|- prod_ j e. A B = prod_ k e. 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 2336	. 2 |- F/_ k A
3		nfcv 2336	. 2 |- F/_ j A
4		nfcv 2336	. 2 |- F/_ k B
5		nfcv 2336	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvprod 11704	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1364   prod_cprod 11696

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-recs 6360  df-frec 6446  df-seqfrec 10522  df-proddc 11697

This theorem is referenced by:  eulerthlemth  12373