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Theorem prodeq1i 29025
Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1i.1  |-  A  =  B
Assertion
Ref Expression
prodeq1i  |-  prod_ k  e.  A  C  =  prod_ k  e.  B  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 29016 . 2  |-  ( A  =  B  ->  prod_ k  e.  A  C  = 
prod_ k  e.  B  C )
31, 2ax-mp 5 1  |-  prod_ k  e.  A  C  =  prod_ k  e.  B  C
Colors of variables: wff setvar class
Syntax hints:    = wceq 1383   prod_cprod 29012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-recs 7044  df-rdg 7078  df-seq 12087  df-prod 29013
This theorem is referenced by:  prodeq12i  29027  fprodxp  29087  risefac0  29124  fallfacfwd  29133
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