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Theorem prodeq1d 15964
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 15951 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2syl 18 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1563  cprod 15947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-xp 5658  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-iota 6481  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-seq 14029  df-prod 15948
This theorem is referenced by:  prodeq12dv  15970  prodeq12rdv  15971  fprodf1o  15990  prodss  15991  fprod1  16007  fprodp1  16013  fprodfac  16017  fprodabs  16018  fprod2d  16025  fprodcom2  16028  risefacval  16052  fallfacval  16053  risefacval2  16054  fallfacval2  16055  risefacp1  16073  fallfacp1  16074  fallfacval4  16087  fprodefsum  16139  prmoval  17083  prmop1  17088  prmgapprmo  17112  gausslemma2dlem4  27491  breprexplema  34934  breprexplemc  34936  breprexp  34937  circlemethhgt  34947  bcprod  36101  aks4d1p1  42705  dvmptfprodlem  46516  dvmptfprod  46517  ovnval  47113  hoiprodp1  47160  hoidmv1le  47166  hspmbllem1  47198  fmtnorec2  48150
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