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Theorem prodeq1i 14692
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 14683 . 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 1523  cprod 14679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seq 12842  df-prod 14680
This theorem is referenced by:  prodeq12i  14694  fprodxp  14756  risefac0  14802  fallfacfwd  14811  prmo0  15787  breprexp  30839  etransclem31  40800  etransclem35  40804  hoidmv1le  41129  fmtnorec2  41780
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