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Theorem prodeq1d 14490
 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 14478 . 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 1475  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-prod 14475 This theorem is referenced by:  prodeq12dv  14495  prodeq12rdv  14496  fprodf1o  14515  prodss  14516  fprod1  14532  fprodp1  14538  fprodfac  14542  fprodabs  14543  fprod2d  14550  fprodcom2  14553  fprodcom2OLD  14554  risefacval  14578  fallfacval  14579  risefacval2  14580  fallfacval2  14581  risefacp1  14599  fallfacp1  14600  fallfacval4  14613  fprodefsum  14664  prmoval  15575  prmop1  15580  prmgapprmo  15604  gausslemma2dlem4  24894  bcprod  30877  dvmptfprodlem  38834  dvmptfprod  38835  ovnval  39431  hoiprodp1  39478  hoidmv1le  39484  hspmbllem1  39516  fmtnorec2  39993
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