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Theorem r19.27mv 3511
Description: Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.27mv  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem r19.27mv
StepHypRef Expression
1 nfv 1521 . 2  |-  F/ x ps
21r19.27m 3510 1  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1485    e. wcel 2141   A.wral 2448
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-ral 2453
This theorem is referenced by:  bezoutlembi  11960
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