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Theorem feq23 5394
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 5392 . 2  |-  ( A  =  C  ->  ( F : A --> B  <->  F : C
--> B ) )
2 feq3 5393 . 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 1632   -->wf 5267
This theorem is referenced by:  feq23i  5401  ismgm  21003  ismndo2  21028  rngomndo  21104  nvof1o  23052  feq123  25171  islatalg  25286  seff  27641
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-fn 5274  df-f 5275
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