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Theorem mpt2eq3ia 5706
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpt2eq3ia.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
Assertion
Ref Expression
mpt2eq3ia  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )

Proof of Theorem mpt2eq3ia
StepHypRef Expression
1 mpt2eq3ia.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
213adant1 961 . . 3  |-  ( ( T.  /\  x  e.  A  /\  y  e.  B )  ->  C  =  D )
32mpt2eq3dva 5705 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A , 
y  e.  B  |->  D ) )
43mptru 1298 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 102    = wceq 1289   T. wtru 1290    e. wcel 1438    |-> cmpt2 5646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-oprab 5648  df-mpt2 5649
This theorem is referenced by:  oprab2co  5975  genpdf  7057  dfioo2  9382  iseqvalcbv  9860
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