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Theorem funeqd 5609
Description: Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
Hypothesis
Ref Expression
funeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
funeqd  |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )

Proof of Theorem funeqd
StepHypRef Expression
1 funeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 funeq 5607 . 2  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
31, 2syl 16 1  |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-in 3483  df-ss 3490  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-fun 5590
This theorem is referenced by:  funopg  5620  funsng  5634  f1eq1  5776  funcnvuni  6737  shftfn  12869  isstruct2  14499  strle1  14586  monfval  14988  ismon  14989  monpropd  14993  isepi  14996  isfth  15141  lubfun  15467  glbfun  15480  acsficl2d  15663  frlmphl  18607  eengbas  23988  ebtwntg  23989  ecgrtg  23990  elntg  23991  istrl  24243  ispth  24274  isspth  24275  0spth  24277  1pthonlem1  24295  constr2spthlem1  24300  2pthlem1  24301  constr2pth  24307  constr3pthlem2  24360  ajfun  25480  sitgf  27957  fperdvper  31276  dfateq12d  31709  afvres  31752  usgra2pthspth  31846
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