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Theorem mpteq2dv 4019
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
mpteq2dv.1  |-  ( ph  ->  B  =  C )
Assertion
Ref Expression
mpteq2dv  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)

Proof of Theorem mpteq2dv
StepHypRef Expression
1 mpteq2dv.1 . . 3  |-  ( ph  ->  B  =  C )
21adantr 274 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
32mpteq2dva 4018 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480    |-> cmpt 3989
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-opab 3990  df-mpt 3991
This theorem is referenced by:  ofeq  5984  rdgeq1  6268  rdgeq2  6269  omv  6351  oeiv  6352  0tonninf  10224  1tonninf  10225  iseqf1olemjpcl  10280  iseqf1olemqpcl  10281  iseqf1olemfvp  10282  seq3f1olemqsum  10285  seq3f1olemp  10287  summodc  11164  zsumdc  11165  fsum3  11168  prodeq2w  11337  prodmodc  11359  sloteq  11978  cnprcl2k  12389  fsumcncntop  12739  expcncf  12775  dvexp  12858  dvexp2  12859  peano4nninf  13261  peano3nninf  13262  nninfalllem1  13264  nninfsellemdc  13267  nninfsellemeq  13271  nninfsellemqall  13272  nninfsellemeqinf  13273  nninfomni  13276
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