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Theorem feq1d 5368
Description: Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
Hypothesis
Ref Expression
feq1d.1  |-  ( ph  ->  F  =  G )
Assertion
Ref Expression
feq1d  |-  ( ph  ->  ( F : A --> B 
<->  G : A --> B ) )

Proof of Theorem feq1d
StepHypRef Expression
1 feq1d.1 . 2  |-  ( ph  ->  F  =  G )
2 feq1 5364 . 2  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
31, 2syl 14 1  |-  ( ph  ->  ( F : A --> B 
<->  G : A --> B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364   -->wf 5228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-fun 5234  df-fn 5235  df-f 5236
This theorem is referenced by:  feq12d  5371  fco2  5398  fssres2  5409  fresin  5410  fmpt3d  5689  fmptco  5699  fressnfv  5720  off  6114  caofinvl  6124  f2ndf  6246  eroprf  6649  pmresg  6697  pw2f1odclem  6857  fseq1p1m1  10119  mgmplusf  12835  mgmb1mgm1  12837  grpsubf  13016  lmodscaf  13619  lmbr  14150  blfps  14346  blf  14347  dvmptclx  14617  lgsfcl3  14860
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