Theorem cbvprodi 12305

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 2392	. 2 |- F/_ k A
3		nfcv 2392	. 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 12303	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1402   F/_wnfc 2379   prod_cprod 12295

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-recs 6566  df-frec 6652  df-seqfrec 10863  df-proddc 12296

This theorem is referenced by:  prodfct  12332  prodsnf  12337  fprodm1s  12346  fprodp1s  12347  prodsns  12348  fprodcllemf  12358  fprod2dlemstep  12367  fprodcom2fi  12371  fproddivapf  12376  fprodsplitf  12377