Theorem cbvprodv 12181

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 2375	. 2 |- F/_ k A
3		nfcv 2375	. 2 |- F/_ j A
4		nfcv 2375	. 2 |- F/_ k B
5		nfcv 2375	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvprod 12180	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1398   prod_cprod 12172

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-frec 6600  df-seqfrec 10754  df-proddc 12173

This theorem is referenced by:  eulerthlemth  12865