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Theorem feq123 25068
Description: Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
Assertion
Ref Expression
feq123  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : C --> D ) )

Proof of Theorem feq123
StepHypRef Expression
1 feq1 5375 . . 3  |-  ( F  =  G  ->  ( F : A --> B  <->  G : A
--> B ) )
213ad2ant1 976 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : A --> B ) )
3 feq23 5378 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( G : A --> B 
<->  G : C --> D ) )
433adant1 973 . 2  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( G : A --> B 
<->  G : C --> D ) )
52, 4bitrd 244 1  |-  ( ( F  =  G  /\  A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  G : C --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623   -->wf 5251
This theorem is referenced by:  fprg  25133  vecval3b  25452  vri  25492  pgapspf  26052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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