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Theorem prodeq1d 15865
Description: Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
prodeq1d (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hints:   𝜑(𝑘)   𝐶(𝑘)

Proof of Theorem prodeq1d
StepHypRef Expression
1 prodeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 prodeq1 15853 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2syl 17 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cprod 15849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-seq 13967  df-prod 15850
This theorem is referenced by:  prodeq12dv  15870  prodeq12rdv  15871  fprodf1o  15890  prodss  15891  fprod1  15907  fprodp1  15913  fprodfac  15917  fprodabs  15918  fprod2d  15925  fprodcom2  15928  risefacval  15952  fallfacval  15953  risefacval2  15954  fallfacval2  15955  risefacp1  15973  fallfacp1  15974  fallfacval4  15987  fprodefsum  16038  prmoval  16966  prmop1  16971  prmgapprmo  16995  gausslemma2dlem4  26872  breprexplema  33642  breprexplemc  33644  breprexp  33645  circlemethhgt  33655  bcprod  34708  aks4d1p1  40941  dvmptfprodlem  44660  dvmptfprod  44661  ovnval  45257  hoiprodp1  45304  hoidmv1le  45310  hspmbllem1  45342  fmtnorec2  46211
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