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Theorem funeqd 5052
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 5050 . 2  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
31, 2syl 14 1  |-  ( ph  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290   Fun wfun 5024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-in 3008  df-ss 3015  df-br 3854  df-opab 3908  df-rel 4461  df-cnv 4462  df-co 4463  df-fun 5032
This theorem is referenced by:  funopg  5063  funsng  5075  funcnvuni  5098  f1eq1  5226  frecuzrdgtclt  9891  shftfn  10321  isstruct2im  11567  isstruct2r  11568  structfung  11574  setsfun  11592  setsfun0  11593
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