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Theorem r19.3rm 3446
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
Hypothesis
Ref Expression
r19.3rm.1  |-  F/ x ph
Assertion
Ref Expression
r19.3rm  |-  ( E. y  y  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem r19.3rm
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 eleq1 2200 . . 3  |-  ( a  =  y  ->  (
a  e.  A  <->  y  e.  A ) )
21cbvexv 1890 . 2  |-  ( E. a  a  e.  A  <->  E. y  y  e.  A
)
3 eleq1 2200 . . . 4  |-  ( a  =  x  ->  (
a  e.  A  <->  x  e.  A ) )
43cbvexv 1890 . . 3  |-  ( E. a  a  e.  A  <->  E. x  x  e.  A
)
5 biimt 240 . . . 4  |-  ( E. x  x  e.  A  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
6 df-ral 2419 . . . . 5  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
7 r19.3rm.1 . . . . . 6  |-  F/ x ph
8719.23 1656 . . . . 5  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
96, 8bitri 183 . . . 4  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
105, 9syl6bbr 197 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
114, 10sylbi 120 . 2  |-  ( E. a  a  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
122, 11sylbir 134 1  |-  ( E. y  y  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329   F/wnf 1436   E.wex 1468    e. wcel 1480   A.wral 2414
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-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2130  df-clel 2133  df-ral 2419
This theorem is referenced by:  r19.28m  3447  r19.3rmv  3448  r19.27m  3453  indstr  9381
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