Theorem cbvprodv 14646

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 2764	. 2 |- F/_ k A
3		nfcv 2764	. 2 |- F/_ j A
4		nfcv 2764	. 2 |- F/_ k B
5		nfcv 2764	. 2 |- F/_ j C
6	1, 2, 3, 4, 5	cbvprod 14645	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1483   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seq 12802  df-prod 14636

This theorem is referenced by:  breprexp  30711  mccl  39830  dvnprodlem3  40163  etransclem6  40457  etransclem37  40488  etransclem46  40497  ovnsubadd  40786  hoidmv1le  40808  hoidmvle  40814  hspmbl  40843  ovnovollem3  40872  vonn0ioo  40901  vonn0icc  40902