Theorem cbvprodi 11523

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 2312	. 2 |- F/_ k A
3		nfcv 2312	. 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 11521	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1348   F/_wnfc 2299   prod_cprod 11513

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-seqfrec 10402  df-proddc 11514

This theorem is referenced by:  prodfct  11550  prodsnf  11555  fprodm1s  11564  fprodp1s  11565  prodsns  11566  fprodcllemf  11576  fprod2dlemstep  11585  fprodcom2fi  11589  fproddivapf  11594  fprodsplitf  11595