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Theorem prodeq1i 11511
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 11503 . 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 set class
Syntax hints:    = wceq 1348   prod_cprod 11500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3526  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-cnv 4617  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-recs 6281  df-frec 6367  df-seqfrec 10389  df-proddc 11501
This theorem is referenced by:  prodeq12i  11513  fprodfac  11565  fprodxp  11574
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