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Theorem r19.3rm 3338
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 2116 . . 3  |-  ( a  =  y  ->  (
a  e.  A  <->  y  e.  A ) )
21cbvexv 1811 . 2  |-  ( E. a  a  e.  A  <->  E. y  y  e.  A
)
3 eleq1 2116 . . . 4  |-  ( a  =  x  ->  (
a  e.  A  <->  x  e.  A ) )
43cbvexv 1811 . . 3  |-  ( E. a  a  e.  A  <->  E. x  x  e.  A
)
5 biimt 234 . . . 4  |-  ( E. x  x  e.  A  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
6 df-ral 2328 . . . . 5  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
7 r19.3rm.1 . . . . . 6  |-  F/ x ph
8719.23 1584 . . . . 5  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
96, 8bitri 177 . . . 4  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
105, 9syl6bbr 191 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
114, 10sylbi 118 . 2  |-  ( E. a  a  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
122, 11sylbir 129 1  |-  ( E. y  y  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102   A.wal 1257   F/wnf 1365   E.wex 1397    e. wcel 1409   A.wral 2323
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-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052  df-ral 2328
This theorem is referenced by:  r19.28m  3339  r19.3rmv  3340  r19.27m  3344  indstr  8632
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