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Theorem feq12d 5719
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 5716 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : A --> C ) )
3 feq12d.2 . . 3  |-  ( ph  ->  A  =  B )
43feq2d 5717 . 2  |-  ( ph  ->  ( G : A --> C 
<->  G : B --> C ) )
52, 4bitrd 253 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   -->wf 5583
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-3an 975  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-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5589  df-fn 5590  df-f 5591
This theorem is referenced by:  feq123d  5720  fprg  6069  smoeq  7021  oif  7954  catcisolem  15290  hofcl  15385  dmdprd  16829  dpjf  16905  pjf2  18528  mat1dimmul  18761  lmbr2  19542  lmff  19584  dfac14  19870  lmmbr2  21449  lmcau  21502  perfdvf  22058  dvnfre  22106  dvle  22159  dvfsumle  22173  dvfsumge  22174  dvmptrecl  22176  uhgrac  23997  isumgra  24007  iswlk  24212  istrl  24231  constr1trl  24282  constr3trllem1  24342  eupap1  24668  resf1o  27241  ismeas  27826  mbfresfi  29654  sdclem1  29855  dfac21  30632  fourierdlem74  31497  fourierdlem103  31526  fourierdlem104  31527  uhg0e  31859  uhgrepe  31861  uhgres  31862
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