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Theorem prodeq1d 15764
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 15752 . 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 1541  cprod 15748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-iota 6445  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-seq 13861  df-prod 15749
This theorem is referenced by:  prodeq12dv  15769  prodeq12rdv  15770  fprodf1o  15789  prodss  15790  fprod1  15806  fprodp1  15812  fprodfac  15816  fprodabs  15817  fprod2d  15824  fprodcom2  15827  risefacval  15851  fallfacval  15852  risefacval2  15853  fallfacval2  15854  risefacp1  15872  fallfacp1  15873  fallfacval4  15886  fprodefsum  15937  prmoval  16865  prmop1  16870  prmgapprmo  16894  gausslemma2dlem4  26669  breprexplema  33055  breprexplemc  33057  breprexp  33058  circlemethhgt  33068  bcprod  34127  aks4d1p1  40471  dvmptfprodlem  44086  dvmptfprod  44087  ovnval  44683  hoiprodp1  44730  hoidmv1le  44736  hspmbllem1  44768  fmtnorec2  45636
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