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Theorem feq123d 5524
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 5523 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
4 feq123d.3 . . 3  |-  ( ph  ->  C  =  D )
5 feq3 5519 . . 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 1649   -->wf 5391
This theorem is referenced by:  feq123  5525  feq23d  5529  fprg  5855  evlfcl  14247  yonedalem3a  14299  yonedalem4c  14302  yonedalem3b  14304  yonedainv  14306  iscau  19101  isuhgra  21206  uhgraeq12d  21210  constr3trllem3  21488  isrngo  21815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-fun 5397  df-fn 5398  df-f 5399
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