Theorem cbvprodi 11725

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 2339	. 2 |- F/_ k A
3		nfcv 2339	. 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 11723	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1364   F/_wnfc 2326   prod_cprod 11715

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-seqfrec 10540  df-proddc 11716

This theorem is referenced by:  prodfct  11752  prodsnf  11757  fprodm1s  11766  fprodp1s  11767  prodsns  11768  fprodcllemf  11778  fprod2dlemstep  11787  fprodcom2fi  11791  fproddivapf  11796  fprodsplitf  11797