Theorem cbvprodv 11331

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 2281	. 2 |- F/_ k A
3		nfcv 2281	. 2 |- F/_ j A
4		nfcv 2281	. 2 |- F/_ k B
5		nfcv 2281	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvprod 11330	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1331   prod_cprod 11322

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10222  df-proddc 11323

This theorem is referenced by: (None)