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Theorem rabsnif 4032
Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)
Hypothesis
Ref Expression
rabsnif.f  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rabsnif  |-  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem rabsnif
StepHypRef Expression
1 rabsnifsb 4031 . . 3  |-  { x  e.  { A }  |  ph }  =  if (
[. A  /  x ]. ph ,  { A } ,  (/) )
2 rabsnif.f . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 3288 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  ps ) )
43ifbid 3894 . . 3  |-  ( A  e.  _V  ->  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  if ( ps ,  { A } ,  (/) ) )
51, 4syl5eq 2517 . 2  |-  ( A  e.  _V  ->  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) ) )
6 rab0 3756 . . . 4  |-  { x  e.  (/)  |  ph }  =  (/)
7 ifid 3909 . . . 4  |-  if ( ps ,  (/) ,  (/) )  =  (/)
86, 7eqtr4i 2496 . . 3  |-  { x  e.  (/)  |  ph }  =  if ( ps ,  (/)
,  (/) )
9 snprc 4027 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
109biimpi 199 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
11 rabeq 3024 . . . 4  |-  ( { A }  =  (/)  ->  { x  e.  { A }  |  ph }  =  { x  e.  (/)  | 
ph } )
1210, 11syl 17 . . 3  |-  ( -.  A  e.  _V  ->  { x  e.  { A }  |  ph }  =  { x  e.  (/)  |  ph } )
1310ifeq1d 3890 . . 3  |-  ( -.  A  e.  _V  ->  if ( ps ,  { A } ,  (/) )  =  if ( ps ,  (/)
,  (/) ) )
148, 12, 133eqtr4a 2531 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) ) )
155, 14pm2.61i 169 1  |-  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   [.wsbc 3255   (/)c0 3722   ifcif 3872   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-nul 3723  df-if 3873  df-sn 3960
This theorem is referenced by:  suppsnop  6947  m1detdiag  19699  1loopgrvd2  39725  1hevtxdg1  39728  1egrvtxdg1  39732
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