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Theorem funbreq 29176
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 29175 . . 3  |-  ( Fun 
F  ->  ( ( A F B  /\  A F C )  ->  B  =  C ) )
54expdimp 437 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  ->  B  =  C ) )
6 breq2 4441 . . . 4  |-  ( B  =  C  ->  ( A F B  <->  A F C ) )
76biimpcd 224 . . 3  |-  ( A F B  ->  ( B  =  C  ->  A F C ) )
87adantl 466 . 2  |-  ( ( Fun  F  /\  A F B )  ->  ( B  =  C  ->  A F C ) )
95, 8impbid 191 1  |-  ( ( Fun  F  /\  A F B )  ->  ( A F C  <->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437   Fun wfun 5572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-id 4785  df-cnv 4997  df-co 4998  df-fun 5580
This theorem is referenced by: (None)
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