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Theorem rabsnt 3502
Description: Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
rabsnt.1  |-  B  e. 
_V
rabsnt.2  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rabsnt  |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rabsnt
StepHypRef Expression
1 rabsnt.1 . . . 4  |-  B  e. 
_V
21snid 3460 . . 3  |-  B  e. 
{ B }
3 id 19 . . 3  |-  ( { x  e.  A  |  ph }  =  { B }  ->  { x  e.  A  |  ph }  =  { B } )
42, 3syl5eleqr 2174 . 2  |-  ( { x  e.  A  |  ph }  =  { B }  ->  B  e.  {
x  e.  A  |  ph } )
5 rabsnt.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
65elrab 2762 . . 3  |-  ( B  e.  { x  e.  A  |  ph }  <->  ( B  e.  A  /\  ps ) )
76simprbi 269 . 2  |-  ( B  e.  { x  e.  A  |  ph }  ->  ps )
84, 7syl 14 1  |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287    e. wcel 1436   {crab 2359   _Vcvv 2615   {csn 3431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2617  df-sn 3437
This theorem is referenced by:  ontr2exmid  4316  onsucsssucexmid  4318  ordsoexmid  4353  unfiexmid  6582
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