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Theorem feq23 5571
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 5569 . 2  |-  ( A  =  C  ->  ( F : A --> B  <->  F : C
--> B ) )
2 feq3 5570 . 2  |-  ( B  =  D  ->  ( F : C --> B  <->  F : C
--> D ) )
31, 2sylan9bb 681 1  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652   -->wf 5442
This theorem is referenced by:  feq23i  5579  ismgm  21900  ismndo2  21925  rngomndo  22001  seff  27506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326  df-fn 5449  df-f 5450
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