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Theorem cbvprodi 11570
Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
cbvprodi.1 𝑘𝐵
cbvprodi.2 𝑗𝐶
cbvprodi.3 (𝑗 = 𝑘𝐵 = 𝐶)
Assertion
Ref Expression
cbvprodi 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Distinct variable group:   𝑗,𝑘,𝐴
Allowed substitution hints:   𝐵(𝑗,𝑘)   𝐶(𝑗,𝑘)

Proof of Theorem cbvprodi
StepHypRef Expression
1 cbvprodi.3 . 2 (𝑗 = 𝑘𝐵 = 𝐶)
2 nfcv 2319 . 2 𝑘𝐴
3 nfcv 2319 . 2 𝑗𝐴
4 cbvprodi.1 . 2 𝑘𝐵
5 cbvprodi.2 . 2 𝑗𝐶
61, 2, 3, 4, 5cbvprod 11568 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wnfc 2306  cprod 11560
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-recs 6308  df-frec 6394  df-seqfrec 10448  df-proddc 11561
This theorem is referenced by:  prodfct  11597  prodsnf  11602  fprodm1s  11611  fprodp1s  11612  prodsns  11613  fprodcllemf  11623  fprod2dlemstep  11632  fprodcom2fi  11636  fproddivapf  11641  fprodsplitf  11642
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