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Theorem r19.27m 3344
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 3338 . . 3  |-  ( E. x  x  e.  A  ->  ( ps  <->  A. x  e.  A  ps )
)
32anbi2d 445 . 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 2458 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4syl6rbbr 192 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 101    <-> wb 102   F/wnf 1365   E.wex 1397    e. wcel 1409   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052  df-ral 2328
This theorem is referenced by:  r19.27mv  3345  raaanlem  3354
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