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Theorem rabn0m 3273
Description: Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
Assertion
Ref Expression
rabn0m  |-  ( E. y  y  e.  {
x  e.  A  |  ph }  <->  E. x  e.  A  ph )
Distinct variable groups:    x, y    y, A    ph, y
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem rabn0m
StepHypRef Expression
1 df-rex 2329 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 rabid 2502 . . 3  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
32exbii 1512 . 2  |-  ( E. x  x  e.  {
x  e.  A  |  ph }  <->  E. x ( x  e.  A  /\  ph ) )
4 nfv 1437 . . 3  |-  F/ y  x  e.  { x  e.  A  |  ph }
5 df-rab 2332 . . . . 5  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65eleq2i 2120 . . . 4  |-  ( y  e.  { x  e.  A  |  ph }  <->  y  e.  { x  |  ( x  e.  A  /\  ph ) } )
7 nfsab1 2046 . . . 4  |-  F/ x  y  e.  { x  |  ( x  e.  A  /\  ph ) }
86, 7nfxfr 1379 . . 3  |-  F/ x  y  e.  { x  e.  A  |  ph }
9 eleq1 2116 . . 3  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  ph }  <->  y  e.  { x  e.  A  |  ph }
) )
104, 8, 9cbvex 1655 . 2  |-  ( E. x  x  e.  {
x  e.  A  |  ph }  <->  E. y  y  e. 
{ x  e.  A  |  ph } )
111, 3, 103bitr2ri 202 1  |-  ( E. y  y  e.  {
x  e.  A  |  ph }  <->  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   E.wex 1397    e. wcel 1409   {cab 2042   E.wrex 2324   {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rex 2329  df-rab 2332
This theorem is referenced by:  exss  3991
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