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Theorem r19.9rmv 3514
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.9rmv  |-  ( E. y  y  e.  A  ->  ( ph  <->  E. x  e.  A  ph ) )
Distinct variable groups:    x, A    y, A    ph, x
Allowed substitution hint:    ph( y)

Proof of Theorem r19.9rmv
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eleq1 2240 . . 3  |-  ( a  =  y  ->  (
a  e.  A  <->  y  e.  A ) )
21cbvexv 1918 . 2  |-  ( E. a  a  e.  A  <->  E. y  y  e.  A
)
3 eleq1 2240 . . . 4  |-  ( a  =  x  ->  (
a  e.  A  <->  x  e.  A ) )
43cbvexv 1918 . . 3  |-  ( E. a  a  e.  A  <->  E. x  x  e.  A
)
5 df-rex 2461 . . . . 5  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
6 19.41v 1902 . . . . 5  |-  ( E. x ( x  e.  A  /\  ph )  <->  ( E. x  x  e.  A  /\  ph )
)
75, 6bitri 184 . . . 4  |-  ( E. x  e.  A  ph  <->  ( E. x  x  e.  A  /\  ph )
)
87baibr 920 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  E. x  e.  A  ph ) )
94, 8sylbi 121 . 2  |-  ( E. a  a  e.  A  ->  ( ph  <->  E. x  e.  A  ph ) )
102, 9sylbir 135 1  |-  ( E. y  y  e.  A  ->  ( ph  <->  E. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   E.wex 1492    e. wcel 2148   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-rex 2461
This theorem is referenced by:  r19.45mv  3516  r19.44mv  3517  iunconstm  3894  fconstfvm  5733  frecabcl  6397  ltexprlemloc  7603  lcmgcdlem  12069  dvdsr02  13205
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