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Theorem prodeq1 14478
 Description: Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
Assertion
Ref Expression
prodeq1 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1
StepHypRef Expression
1 nfcv 2751 . 2 𝑘𝐴
2 nfcv 2751 . 2 𝑘𝐵
31, 2prodeq1f 14477 1 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-prod 14475 This theorem is referenced by:  prodeq1i  14487  prodeq1d  14490  prod1  14513  fprodf1o  14515  fprodss  14517  fprodcllem  14520  fprodmul  14529  fproddiv  14530  fprodconst  14547  fprodn0  14548  fprod2d  14550  fprodmodd  14567  coprmprod  15213  coprmproddvds  15215  fprodexp  38661  fprodabs2  38662  mccl  38665  fprodcn  38667  fprodcncf  38787  dvmptfprod  38835  dvnprodlem3  38838  hoidmvval  39467  ovnhoi  39493  hspmbllem2  39517  fmtnorec2  39993
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