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Theorem mpteq2da 4121
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 2296 . . 3  |-  A  =  A
21ax-gen 1536 . 2  |-  A. x  A  =  A
3 mpteq2da.1 . . 3  |-  F/ x ph
4 mpteq2da.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
54ex 423 . . 3  |-  ( ph  ->  ( x  e.  A  ->  B  =  C ) )
63, 5ralrimi 2637 . 2  |-  ( ph  ->  A. x  e.  A  B  =  C )
7 mpteq12f 4112 . 2  |-  ( ( A. x  A  =  A  /\  A. x  e.  A  B  =  C )  ->  (
x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
82, 6, 7sylancr 644 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   A.wral 2556    e. cmpt 4093
This theorem is referenced by:  mpteq2dva  4122  sumeq1f  12177  sumeq2ii  12182  xkocnv  17521  offval2f  23248  esumf1o  23444  cprodeq1f  24130  cprodeq2ii  24135  cnegvex2  25763  rnegvex2  25764  mzpsubmpt  26924  mzpexpmpt  26926  refsum2cnlem1  27811  fmuldfeqlem1  27815  stoweidlem2  27854  stoweidlem4  27856  stoweidlem6  27858  stoweidlem8  27860  stoweidlem17  27869  stoweidlem19  27871  stoweidlem20  27872  stoweidlem21  27873  stoweidlem22  27874  stoweidlem23  27875  stoweidlem32  27884  stoweidlem36  27888  stoweidlem40  27892  stoweidlem41  27893  stoweidlem47  27899  stirlinglem15  27940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ral 2561  df-opab 4094  df-mpt 4095
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