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Theorem r19.28m 3338
Description: Restricted quantifier version of Theorem 19.28 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
r19.28m.1  |-  F/ x ph
Assertion
Ref Expression
r19.28m  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( ph  /\  A. x  e.  A  ps ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem r19.28m
StepHypRef Expression
1 r19.28m.1 . . . 4  |-  F/ x ph
21r19.3rm 3337 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
32anbi1d 453 . 2  |-  ( E. x  x  e.  A  ->  ( ( ph  /\  A. x  e.  A  ps ) 
<->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
4 r19.26 2486 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4syl6rbbr 197 1  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( ph  /\  A. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   F/wnf 1390   E.wex 1422    e. wcel 1434   A.wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2075  df-clel 2078  df-ral 2354
This theorem is referenced by:  r19.28mv  3341  raaanlem  3354
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