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Theorem prodeq1 25235
Description: Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
Assertion
Ref Expression
prodeq1  |-  ( A  =  B  ->  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 prodeq1
StepHypRef Expression
1 nfcv 2572 . 2  |-  F/_ k A
2 nfcv 2572 . 2  |-  F/_ k B
31, 2prodeq1f 25234 1  |-  ( A  =  B  ->  prod_ k  e.  A C  = 
prod_ k  e.  B C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   prod_cprod 25231
This theorem is referenced by:  prodeq1i  25244  prodeq1d  25247  prod1  25270  fprodf1o  25272  fprodss  25274  fprodcllem  25277  fprodmul  25284  fproddiv  25285  fprodconst  25302  fprodn0  25303  fprod2d  25305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-seq 11324  df-prod 25232
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