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Theorem rabsnt 3704
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 3667 . . 3  |-  B  e. 
{ B }
3 id 19 . . 3  |-  ( { x  e.  A  |  ph }  =  { B }  ->  { x  e.  A  |  ph }  =  { B } )
42, 3syl5eleqr 2370 . 2  |-  ( { x  e.  A  |  ph }  =  { B }  ->  B  e.  {
x  e.  A  |  ph } )
5 rabsnt.2 . . . 4  |-  ( x  =  B  ->  ( ph 
<->  ps ) )
65elrab 2923 . . 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-sn 3646
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