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Theorem prodeq1d 25200
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 25188 . 2  |-  ( A  =  B  ->  prod_ k  e.  A C  = 
prod_ k  e.  B C )
31, 2syl 16 1  |-  ( ph  ->  prod_ k  e.  A C  =  prod_ k  e.  B C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   prod_cprod 25184
This theorem is referenced by:  prodeq12dv  25205  prodeq12rdv  25206  fprodf1o  25225  prodss  25226  fprod1  25240  fprodp1  25245  fprodfac  25249  fprodabs  25250  fprodefsum  25251  fprod2d  25258  fprodcom2  25261  risefacval  25277  fallfacval  25278  risefacval2  25279  fallfacval2  25280  risefacp1  25297  fallfacp1  25298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-seq 11279  df-prod 25185
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