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Theorem feq123 25171
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 5391 . . 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 5394 . . 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 1632   -->wf 5267
This theorem is referenced by:  fprg  25236  vecval3b  25555  vri  25595  pgapspf  26155
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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