Theorem cbvprodi 11986

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 2350	. 2 |- F/_ k A
3		nfcv 2350	. 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 11984	1 |- prod_ j e. A B = prod_ k e. A C

Syntax hints:   -> wi 4   = wceq 1373   F/_wnfc 2337   prod_cprod 11976

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-if 3580  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-recs 6414  df-frec 6500  df-seqfrec 10630  df-proddc 11977

This theorem is referenced by:  prodfct  12013  prodsnf  12018  fprodm1s  12027  fprodp1s  12028  prodsns  12029  fprodcllemf  12039  fprod2dlemstep  12048  fprodcom2fi  12052  fproddivapf  12057  fprodsplitf  12058