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Theorem rabsnif 4013
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 4012 . . 3  |-  { x  e.  { A }  |  ph }  =  if (
[. A  /  x ]. ph ,  { A } ,  (/) )
2 rabsnif.f . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 3285 . . . 4  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  ps ) )
43ifbid 3879 . . 3  |-  ( A  e.  _V  ->  if ( [. A  /  x ]. ph ,  { A } ,  (/) )  =  if ( ps ,  { A } ,  (/) ) )
51, 4syl5eq 2435 . 2  |-  ( A  e.  _V  ->  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) ) )
6 rab0 3733 . . . 4  |-  { x  e.  (/)  |  ph }  =  (/)
7 ifid 3894 . . . 4  |-  if ( ps ,  (/) ,  (/) )  =  (/)
86, 7eqtr4i 2414 . . 3  |-  { x  e.  (/)  |  ph }  =  if ( ps ,  (/)
,  (/) )
9 snprc 4007 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
109biimpi 194 . . . 4  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
11 rabeq 3028 . . . 4  |-  ( { A }  =  (/)  ->  { x  e.  { A }  |  ph }  =  { x  e.  (/)  | 
ph } )
1210, 11syl 16 . . 3  |-  ( -.  A  e.  _V  ->  { x  e.  { A }  |  ph }  =  { x  e.  (/)  |  ph } )
1310ifeq1d 3875 . . 3  |-  ( -.  A  e.  _V  ->  if ( ps ,  { A } ,  (/) )  =  if ( ps ,  (/)
,  (/) ) )
148, 12, 133eqtr4a 2449 . 2  |-  ( -.  A  e.  _V  ->  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) ) )
155, 14pm2.61i 164 1  |-  { x  e.  { A }  |  ph }  =  if ( ps ,  { A } ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1399    e. wcel 1826   {crab 2736   _Vcvv 3034   [.wsbc 3252   (/)c0 3711   ifcif 3857   {csn 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-nul 3712  df-if 3858  df-sn 3945
This theorem is referenced by:  suppsnop  6831  m1detdiag  19184
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