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Theorem r19.27m 3428
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.)
Hypothesis
Ref Expression
r19.27m.1  |-  F/ x ps
Assertion
Ref Expression
r19.27m  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem r19.27m
StepHypRef Expression
1 r19.27m.1 . . . 4  |-  F/ x ps
21r19.3rm 3421 . . 3  |-  ( E. x  x  e.  A  ->  ( ps  <->  A. x  e.  A  ps )
)
32anbi2d 459 . 2  |-  ( E. x  x  e.  A  ->  ( ( A. x  e.  A  ph  /\  ps ) 
<->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
4 r19.26 2535 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4syl6rbbr 198 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   F/wnf 1421   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:  r19.27mv  3429  raaanlem  3438
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