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Theorem mpteq12dva 3979
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpteq12dv.1  |-  ( ph  ->  A  =  C )
mpteq12dva.2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  D )
Assertion
Ref Expression
mpteq12dva  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Distinct variable group:    ph, x
Allowed substitution hints:    A( x)    B( x)    C( x)    D( x)

Proof of Theorem mpteq12dva
StepHypRef Expression
1 mpteq12dv.1 . . 3  |-  ( ph  ->  A  =  C )
21alrimiv 1830 . 2  |-  ( ph  ->  A. x  A  =  C )
3 mpteq12dva.2 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  =  D )
43ralrimiva 2482 . 2  |-  ( ph  ->  A. x  e.  A  B  =  D )
5 mpteq12f 3978 . 2  |-  ( ( A. x  A  =  C  /\  A. x  e.  A  B  =  D )  ->  (
x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
62, 4, 5syl2anc 408 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393    |-> cmpt 3959
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-opab 3960  df-mpt 3961
This theorem is referenced by:  mpteq12dv  3980
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