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Theorem prodeq1i 15864
Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
prodeq1i 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1i
StepHypRef Expression
1 prodeq1i.1 . 2 𝐴 = 𝐵
2 prodeq1 15855 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2ax-mp 5 1 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cprod 15851
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 2703
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 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seq 13969  df-prod 15852
This theorem is referenced by:  prodeq12i  15866  fprodxp  15928  risefac0  15973  fallfacfwd  15982  prmo0  16971  breprexp  33714  etransclem31  45060  etransclem35  45064  hoidmv1le  45389  fmtnorec2  46290
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