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Theorem feq123d 5398
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 5397 . 2  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> C ) )
4 feq123d.3 . . 3  |-  ( ph  ->  C  =  D )
5 feq3 5393 . . 3  |-  ( C  =  D  ->  ( G : B --> C  <->  G : B
--> D ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( G : B --> C 
<->  G : B --> D ) )
73, 6bitrd 244 1  |-  ( ph  ->  ( F : A --> C 
<->  G : B --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632   -->wf 5267
This theorem is referenced by:  feq23d  5402  yonedalem3a  14064  yonedalem4c  14067  yonedalem3b  14069  yonedainv  14071  constr3trllem3  28398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275
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