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Theorem feq23 5378
Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
feq23  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )

Proof of Theorem feq23
StepHypRef Expression
1 feq2 5376 . 2  |-  ( A  =  C  ->  ( F : A --> B  <->  F : C
--> B ) )
2 feq3 5377 . 2  |-  ( B  =  D  ->  ( F : C --> B  <->  F : C
--> D ) )
31, 2sylan9bb 680 1  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   -->wf 5251
This theorem is referenced by:  feq23i  5385  ismgm  20987  ismndo2  21012  rngomndo  21088  nvof1o  23036  feq123  25068  islatalg  25183  seff  27538
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-fn 5258  df-f 5259
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