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Theorem preqsnd 23953
Description: Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Hypotheses
Ref Expression
preqsnd.1  |-  ( ph  ->  A  e.  _V )
preqsnd.2  |-  ( ph  ->  B  e.  _V )
preqsnd.3  |-  ( ph  ->  C  e.  _V )
Assertion
Ref Expression
preqsnd  |-  ( ph  ->  ( { A ,  B }  =  { C }  <->  ( A  =  C  /\  B  =  C ) ) )

Proof of Theorem preqsnd
StepHypRef Expression
1 preqsnd.1 . 2  |-  ( ph  ->  A  e.  _V )
2 preqsnd.2 . 2  |-  ( ph  ->  B  e.  _V )
3 preqsnd.3 . 2  |-  ( ph  ->  C  e.  _V )
4 dfsn2 3788 . . . 4  |-  { C }  =  { C ,  C }
54eqeq2i 2414 . . 3  |-  ( { A ,  B }  =  { C }  <->  { A ,  B }  =  { C ,  C }
)
6 preq12bg 3937 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( C  e.  _V  /\  C  e.  _V )
)  ->  ( { A ,  B }  =  { C ,  C } 
<->  ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C )
) ) )
7 oridm 501 . . . 4  |-  ( ( ( A  =  C  /\  B  =  C )  \/  ( A  =  C  /\  B  =  C ) )  <->  ( A  =  C  /\  B  =  C ) )
86, 7syl6bb 253 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( C  e.  _V  /\  C  e.  _V )
)  ->  ( { A ,  B }  =  { C ,  C } 
<->  ( A  =  C  /\  B  =  C ) ) )
95, 8syl5bb 249 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( C  e.  _V  /\  C  e.  _V )
)  ->  ( { A ,  B }  =  { C }  <->  ( A  =  C  /\  B  =  C ) ) )
101, 2, 3, 3, 9syl22anc 1185 1  |-  ( ph  ->  ( { A ,  B }  =  { C }  <->  ( A  =  C  /\  B  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   {csn 3774   {cpr 3775
This theorem is referenced by:  disjdifprg  23970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-pr 3781
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