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Theorem feq12d 5735
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 5732 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : A --> C ) )
3 feq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43feq2d 5733 . 2  |-  ( ph  ->  ( G : A --> C 
<->  G : B --> C ) )
52, 4bitrd 256 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   -->wf 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-fun 5603  df-fn 5604  df-f 5605
This theorem is referenced by:  feq123d  5736  fprg  6088  smoeq  7077  oif  8045  catcisolem  15952  hofcl  16095  dmdprd  17565  dpjf  17625  pjf2  19208  mat1dimmul  19432  lmbr2  20206  lmff  20248  dfac14  20564  lmmbr2  22122  lmcau  22175  perfdvf  22735  dvnfre  22783  dvle  22836  dvfsumle  22850  dvfsumge  22851  dvmptrecl  22853  uhgrac  24878  isumgra  24888  iswlk  25093  istrl  25112  constr1trl  25163  constr3trllem1  25223  eupap1  25549  resf1o  28158  ismeas  28860  omsmeas  28984  mbfresfi  31690  sdclem1  31775  dfac21  35629  fourierdlem74  37611  fourierdlem103  37640  fourierdlem104  37641  sge0iunmpt  37793  ismea  37797  isome  37823  uhg0e  38445  uhgrepe  38447  uhgres  38448
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