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Theorem mpteq2da 3874
Description: Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
Hypotheses
Ref Expression
mpteq2da.1  |-  F/ x ph
mpteq2da.2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
mpteq2da  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )

Proof of Theorem mpteq2da
StepHypRef Expression
1 eqid 2056 . . 3  |-  A  =  A
21ax-gen 1354 . 2  |-  A. x  A  =  A
3 mpteq2da.1 . . 3  |-  F/ x ph
4 mpteq2da.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
54ex 112 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
63, 5ralrimi 2407 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
7 mpteq12f 3865 . 2  |-  ( ( A. x  A  =  A  /\  A. x  e.  A  B  =  C )  ->  (
x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
82, 6, 7sylancr 399 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257    = wceq 1259   F/wnf 1365    e. wcel 1409   A.wral 2323    |-> cmpt 3846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-opab 3847  df-mpt 3848
This theorem is referenced by:  mpteq2dva  3875
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