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Theorem funbreq 25395
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 25394 . . 3  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
54expdimp 427 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  ->  B  =  C ) )
6 breq2 4216 . . . 4  |-  ( B  =  C  ->  ( A F B  <->  A F C ) )
76biimpcd 216 . . 3  |-  ( A F B  ->  ( B  =  C  ->  A F C ) )
87adantl 453 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( B  =  C  ->  A F C ) )
95, 8impbid 184 1  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   Fun wfun 5448
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-cnv 4886  df-co 4887  df-fun 5456
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