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Theorem feq23 5714
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 5712 . 2  |-  ( A  =  C  ->  ( F : A --> B  <->  F : C
--> B ) )
2 feq3 5713 . 2  |-  ( B  =  D  ->  ( F : C --> B  <->  F : C
--> D ) )
31, 2sylan9bb 699 1  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F : A --> B 
<->  F : C --> D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   -->wf 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3483  df-ss 3490  df-fn 5589  df-f 5590
This theorem is referenced by:  feq23i  5723  ismgm  24998  ismndo2  25023  rngomndo  25099  seff  30792
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