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Theorem r19.3rmv 3423
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
Assertion
Ref Expression
r19.3rmv  |-  ( E. y  y  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
Distinct variable groups:    x, A    y, A    ph, x
Allowed substitution hint:    ph( y)

Proof of Theorem r19.3rmv
StepHypRef Expression
1 nfv 1493 . 2  |-  F/ x ph
21r19.3rm 3421 1  |-  ( E. y  y  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   E.wex 1453    e. wcel 1465   A.wral 2393
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-ral 2398
This theorem is referenced by:  iinconstm  3792  exmidsssnc  4096  cnvpom  5051  ssfilem  6737  diffitest  6749  inffiexmid  6768  ctssexmid  6992  exmidonfinlem  7017  caucvgsrlemasr  7566  resqrexlemgt0  10760
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