Theorem cbvprodv 12143

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 2373	. 2 |- F/_ k A
3		nfcv 2373	. 2 |- F/_ j A
4		nfcv 2373	. 2 |- F/_ k B
5		nfcv 2373	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvprod 12142	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1397   prod_cprod 12134

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-if 3605  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-cnv 4735  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-recs 6476  df-frec 6562  df-seqfrec 10716  df-proddc 12135

This theorem is referenced by:  eulerthlemth  12827