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Theorem feq123d 5727
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 )
feq123d.3  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
feq123d  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> D ) )

Proof of Theorem feq123d
StepHypRef Expression
1 feq12d.1 . . 3  |-  ( ph  ->  F  =  G )
2 feq12d.2 . . 3  |-  ( ph  ->  A  =  B )
31, 2feq12d 5726 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
4 feq123d.3 . . 3  |-  ( ph  ->  C  =  D )
54feq3d 5725 . 2  |-  ( ph  ->  ( G : B --> C 
<->  G : B --> D ) )
63, 5bitrd 253 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598
This theorem is referenced by:  feq123  5728  feq23d  5732  fprg  6081  csbwrdg  12578  funcestrcsetclem8  15543  funcsetcestrclem8  15558  funcsetcestrclem9  15559  evlfcl  15618  yonedalem3a  15670  yonedalem4c  15673  yonedalem3b  15675  yonedainv  15677  iscau  21841  isuhgra  24425  uhgraeq12d  24434  constr3trllem3  24779  isrngo  25507  sseqf  28528  ismfs  29106  isuhgr  32607  uhgeq12g  32611  uhguhgra  32613  uhgrauhg  32614  funcringcsetcOLD2lem8  32974  funcringcsetclem8OLD  32997
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