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Theorem rabn0m 3390
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 2422 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
2 rabid 2606 . . 3  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
32exbii 1584 . 2  |-  ( E. x  x  e.  {
x  e.  A  |  ph }  <->  E. x ( x  e.  A  /\  ph ) )
4 nfv 1508 . . 3  |-  F/ y  x  e.  { x  e.  A  |  ph }
5 df-rab 2425 . . . . 5  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65eleq2i 2206 . . . 4  |-  ( y  e.  { x  e.  A  |  ph }  <->  y  e.  { x  |  ( x  e.  A  /\  ph ) } )
7 nfsab1 2129 . . . 4  |-  F/ x  y  e.  { x  |  ( x  e.  A  /\  ph ) }
86, 7nfxfr 1450 . . 3  |-  F/ x  y  e.  { x  e.  A  |  ph }
9 eleq1 2202 . . 3  |-  ( x  =  y  ->  (
x  e.  { x  e.  A  |  ph }  <->  y  e.  { x  e.  A  |  ph }
) )
104, 8, 9cbvex 1729 . 2  |-  ( E. x  x  e.  {
x  e.  A  |  ph }  <->  E. y  y  e. 
{ x  e.  A  |  ph } )
111, 3, 103bitr2ri 208 1  |-  ( E. y  y  e.  {
x  e.  A  |  ph }  <->  E. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1468    e. wcel 1480   {cab 2125   E.wrex 2417   {crab 2420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-rex 2422  df-rab 2425
This theorem is referenced by:  exss  4149  cc4f  7091  cc4n  7093
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