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Theorem prodeq1d 15870
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 15858 . 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 1540  cprod 15854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 7415  df-oprab 7416  df-mpo 7417  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-seq 13972  df-prod 15855
This theorem is referenced by:  prodeq12dv  15875  prodeq12rdv  15876  fprodf1o  15895  prodss  15896  fprod1  15912  fprodp1  15918  fprodfac  15922  fprodabs  15923  fprod2d  15930  fprodcom2  15933  risefacval  15957  fallfacval  15958  risefacval2  15959  fallfacval2  15960  risefacp1  15978  fallfacp1  15979  fallfacval4  15992  fprodefsum  16043  prmoval  16971  prmop1  16976  prmgapprmo  17000  gausslemma2dlem4  27105  breprexplema  33937  breprexplemc  33939  breprexp  33940  circlemethhgt  33950  bcprod  35009  aks4d1p1  41248  dvmptfprodlem  44960  dvmptfprod  44961  ovnval  45557  hoiprodp1  45604  hoidmv1le  45610  hspmbllem1  45642  fmtnorec2  46511
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