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Theorem rabsnt 3636
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 3592 . . 3  |-  B  e. 
{ B }
3 id 19 . . 3  |-  ( { x  e.  A  |  ph }  =  { B }  ->  { x  e.  A  |  ph }  =  { B } )
42, 3eleqtrrid 2247 . 2  |-  ( { x  e.  A  |  ph }  =  { B }  ->  B  e.  {
x  e.  A  |  ph } )
5 rabsnt.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
65elrab 2868 . . 3  |-  ( B  e.  { x  e.  A  |  ph }  <->  ( B  e.  A  /\  ps ) )
76simprbi 273 . 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 104    = wceq 1335    e. wcel 2128   {crab 2439   _Vcvv 2712   {csn 3561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rab 2444  df-v 2714  df-sn 3567
This theorem is referenced by:  ontr2exmid  4486  onsucsssucexmid  4488  ordsoexmid  4523  unfiexmid  6864
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