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Theorem rabsnt 3717
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 3680 . . 3  |-  B  e. 
{ B }
3 id 19 . . 3  |-  ( { x  e.  A  |  ph }  =  { B }  ->  { x  e.  A  |  ph }  =  { B } )
42, 3syl5eleqr 2383 . 2  |-  ( { x  e.  A  |  ph }  =  { B }  ->  B  e.  {
x  e.  A  |  ph } )
5 rabsnt.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
65elrab 2936 . . 3  |-  ( B  e.  { x  e.  A  |  ph }  <->  ( B  e.  A  /\  ps ) )
76simprbi 450 . 2  |-  ( B  e.  { x  e.  A  |  ph }  ->  ps )
84, 7syl 15 1  |-  ( { x  e.  A  |  ph }  =  { B }  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sn 3659
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