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Theorem mpoeq3ia 5915
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpoeq3ia.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
Assertion
Ref Expression
mpoeq3ia  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )

Proof of Theorem mpoeq3ia
StepHypRef Expression
1 mpoeq3ia.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
213adant1 1010 . . 3  |-  ( ( T.  /\  x  e.  A  /\  y  e.  B )  ->  C  =  D )
32mpoeq3dva 5914 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A , 
y  e.  B  |->  D ) )
43mptru 1357 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348   T. wtru 1349    e. wcel 2141    e. cmpo 5852
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  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-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-oprab 5854  df-mpo 5855
This theorem is referenced by:  mpodifsnif  5943  mposnif  5944  oprab2co  6194  genpdf  7457  dfioo2  9918  iseqvalcbv  10400  divcnap  13308
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