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Theorem prodeq1d 15133
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 15121 . 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 1508  cprod 15117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-xp 5409  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-iota 6149  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-seq 13183  df-prod 15118
This theorem is referenced by:  prodeq12dv  15138  prodeq12rdv  15139  fprodf1o  15158  prodss  15159  fprod1  15175  fprodp1  15181  fprodfac  15185  fprodabs  15186  fprod2d  15193  fprodcom2  15196  risefacval  15220  fallfacval  15221  risefacval2  15222  fallfacval2  15223  risefacp1  15241  fallfacp1  15242  fallfacval4  15255  fprodefsum  15306  prmoval  16223  prmop1  16228  prmgapprmo  16252  gausslemma2dlem4  25662  breprexplema  31581  breprexplemc  31583  breprexp  31584  circlemethhgt  31594  bcprod  32527  dvmptfprodlem  41693  dvmptfprod  41694  ovnval  42288  hoiprodp1  42335  hoidmv1le  42341  hspmbllem1  42373  fmtnorec2  43107
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