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Theorem rababg 36179
Description: Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)
Assertion
Ref Expression
rababg  |-  ( A. x ( ph  ->  x  e.  A )  <->  { x  e.  A  |  ph }  =  { x  |  ph } )

Proof of Theorem rababg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ancrb 553 . . 3  |-  ( (
ph  ->  x  e.  A
)  <->  ( ph  ->  ( x  e.  A  /\  ph ) ) )
21albii 1691 . 2  |-  ( A. x ( ph  ->  x  e.  A )  <->  A. x
( ph  ->  ( x  e.  A  /\  ph ) ) )
3 nfv 1761 . . 3  |-  F/ y ( ph  ->  (
x  e.  A  /\  ph ) )
4 nfsab1 2441 . . . 4  |-  F/ x  y  e.  { x  |  ph }
5 nfrab1 2971 . . . . 5  |-  F/_ x { x  e.  A  |  ph }
65nfcri 2586 . . . 4  |-  F/ x  y  e.  { x  e.  A  |  ph }
74, 6nfim 2003 . . 3  |-  F/ x
( y  e.  {
x  |  ph }  ->  y  e.  { x  e.  A  |  ph }
)
8 abid 2439 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
9 eleq1 2517 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { x  |  ph }  <->  y  e.  { x  |  ph }
) )
108, 9syl5bbr 263 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  y  e.  { x  |  ph } ) )
11 rabid 2967 . . . . 5  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
12 eleq1 2517 . . . . 5  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  ph }  <->  y  e.  { x  e.  A  |  ph }
) )
1311, 12syl5bbr 263 . . . 4  |-  ( x  =  y  ->  (
( x  e.  A  /\  ph )  <->  y  e.  { x  e.  A  |  ph } ) )
1410, 13imbi12d 322 . . 3  |-  ( x  =  y  ->  (
( ph  ->  ( x  e.  A  /\  ph ) )  <->  ( y  e.  { x  |  ph }  ->  y  e.  {
x  e.  A  |  ph } ) ) )
153, 7, 14cbval 2114 . 2  |-  ( A. x ( ph  ->  ( x  e.  A  /\  ph ) )  <->  A. y
( y  e.  {
x  |  ph }  ->  y  e.  { x  e.  A  |  ph }
) )
16 eqss 3447 . . 3  |-  ( { x  e.  A  |  ph }  =  { x  |  ph }  <->  ( {
x  e.  A  |  ph }  C_  { x  |  ph }  /\  {
x  |  ph }  C_ 
{ x  e.  A  |  ph } ) )
17 rabssab 3516 . . . 4  |-  { x  e.  A  |  ph }  C_ 
{ x  |  ph }
1817biantrur 509 . . 3  |-  ( { x  |  ph }  C_ 
{ x  e.  A  |  ph }  <->  ( {
x  e.  A  |  ph }  C_  { x  |  ph }  /\  {
x  |  ph }  C_ 
{ x  e.  A  |  ph } ) )
19 dfss2 3421 . . 3  |-  ( { x  |  ph }  C_ 
{ x  e.  A  |  ph }  <->  A. y
( y  e.  {
x  |  ph }  ->  y  e.  { x  e.  A  |  ph }
) )
2016, 18, 193bitr2ri 278 . 2  |-  ( A. y ( y  e. 
{ x  |  ph }  ->  y  e.  {
x  e.  A  |  ph } )  <->  { x  e.  A  |  ph }  =  { x  |  ph } )
212, 15, 203bitri 275 1  |-  ( A. x ( ph  ->  x  e.  A )  <->  { x  e.  A  |  ph }  =  { x  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887   {cab 2437   {crab 2741    C_ wss 3404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-in 3411  df-ss 3418
This theorem is referenced by: (None)
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