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Theorem prodeq1d 24548
Description: Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
prodeq1d  |-  ( ph  ->  prod_ k  e.  A C  =  prod_ k  e.  B C )
Distinct variable groups:    A, k    B, k
Allowed substitution hints:    ph( k)    C( k)

Proof of Theorem prodeq1d
StepHypRef Expression
1 prodeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 prodeq1 24536 . 2  |-  ( A  =  B  ->  prod_ k  e.  A C  = 
prod_ k  e.  B C )
31, 2syl 15 1  |-  ( ph  ->  prod_ k  e.  A C  =  prod_ k  e.  B C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   prod_cprod 24532
This theorem is referenced by:  prodeq12dv  24553  prodeq12rdv  24554  fprodf1o  24573  prodss  24574  fprod1  24588  fprodp1  24593  fprodfac  24597  fprodabs  24598  fprodefsum  24599  risefacval  24615  fallfacval  24616  risefacval2  24617  fallfacval2  24618  risefacp1  24634  fallfacp1  24635  gammacvglem1  24651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-seq 11139  df-prod 24533
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