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Theorem funbreq 24127
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 24126 . . 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 4027 . . . 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 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   Fun wfun 5249
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-cnv 4697  df-co 4698  df-fun 5257
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