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Theorem feq12d 5582
Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypotheses
Ref Expression
feq12d.1  |-  ( ph  ->  F  =  G )
feq12d.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
feq12d  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )

Proof of Theorem feq12d
StepHypRef Expression
1 feq12d.1 . . 3  |-  ( ph  ->  F  =  G )
21feq1d 5580 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : A --> C ) )
3 feq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43feq2d 5581 . 2  |-  ( ph  ->  ( G : A --> C 
<->  G : B --> C ) )
52, 4bitrd 245 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   -->wf 5450
This theorem is referenced by:  feq123d  5583  fprg  5915  smoeq  6612  oif  7499  catcisolem  14261  hofcl  14356  dmdprd  15559  dpjf  15615  pjf2  16941  lmbr2  17323  lmff  17365  dfac14  17650  lmmbr2  19212  lmcau  19265  perfdvf  19790  dvnfre  19838  dvle  19891  dvfsumle  19905  dvfsumge  19906  dvmptrecl  19908  isumgra  21350  iswlk  21527  istrl  21537  constr1trl  21588  constr3trllem1  21637  eupap1  21698  ismeas  24553  mbfresfi  26253  sdclem1  26447  dfac21  27141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458
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