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Theorem abn0m 3383
Description: Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
Assertion
Ref Expression
abn0m  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x ph )
Distinct variable groups:    ph, y    x, y
Allowed substitution hint:    ph( x)

Proof of Theorem abn0m
StepHypRef Expression
1 nfv 1508 . . 3  |-  F/ y  x  e.  { x  |  ph }
2 nfsab1 2127 . . 3  |-  F/ x  y  e.  { x  |  ph }
3 eleq1w 2198 . . 3  |-  ( x  =  y  ->  (
x  e.  { x  |  ph }  <->  y  e.  { x  |  ph }
) )
41, 2, 3cbvex 1729 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. y  y  e.  {
x  |  ph }
)
5 abid 2125 . . 3  |-  ( x  e.  { x  | 
ph }  <->  ph )
65exbii 1584 . 2  |-  ( E. x  x  e.  {
x  |  ph }  <->  E. x ph )
74, 6bitr3i 185 1  |-  ( E. y  y  e.  {
x  |  ph }  <->  E. x ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1468    e. wcel 1480   {cab 2123
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
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-clel 2133
This theorem is referenced by:  mapprc  6539
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