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Theorem feq123d 5575
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 5574 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
4 feq123d.3 . . 3  |-  ( ph  ->  C  =  D )
5 feq3 5570 . . 3  |-  ( C  =  D  ->  ( G : B --> C  <->  G : B
--> D ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( G : B --> C 
<->  G : B --> D ) )
73, 6bitrd 245 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   -->wf 5442
This theorem is referenced by:  feq123  5576  feq23d  5580  fprg  5907  evlfcl  14311  yonedalem3a  14363  yonedalem4c  14366  yonedalem3b  14368  yonedainv  14370  iscau  19221  isuhgra  21330  uhgraeq12d  21334  constr3trllem3  21631  isrngo  21958
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450
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