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Theorem funbreq 24198
Description: An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
funbreq.1  |-  A  e. 
_V
funbreq.2  |-  B  e. 
_V
funbreq.3  |-  C  e. 
_V
Assertion
Ref Expression
funbreq  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  <->  B  =  C ) )

Proof of Theorem funbreq
StepHypRef Expression
1 funbreq.1 . . . 4  |-  A  e. 
_V
2 funbreq.2 . . . 4  |-  B  e. 
_V
3 funbreq.3 . . . 4  |-  C  e. 
_V
41, 2, 3fununiq 24197 . . 3  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
54expdimp 426 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  ->  B  =  C ) )
6 breq2 4043 . . . 4  |-  ( B  =  C  ->  ( A F B  <->  A F C ) )
76biimpcd 215 . . 3  |-  ( A F B  ->  ( B  =  C  ->  A F C ) )
87adantl 452 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( B  =  C  ->  A F C ) )
95, 8impbid 183 1  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   Fun wfun 5265
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  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-id 4325  df-cnv 4713  df-co 4714  df-fun 5273
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