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Theorem r19.3rmv 3513
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 1528 . 2  |-  F/ x ph
21r19.3rm 3511 1  |-  ( E. y  y  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   E.wex 1492    e. wcel 2148   A.wral 2455
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-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-cleq 2170  df-clel 2173  df-ral 2460
This theorem is referenced by:  iinconstm  3895  exmidsssnc  4203  cnvpom  5171  ssfilem  6874  diffitest  6886  inffiexmid  6905  ctssexmid  7147  exmidonfinlem  7191  caucvgsrlemasr  7788  resqrexlemgt0  11024
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