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Theorem r19.28m 3483
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 r19.26 2583 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
2 r19.28m.1 . . . 4  |-  F/ x ph
32r19.3rm 3482 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  A. x  e.  A  ph ) )
43anbi1d 461 . 2  |-  ( E. x  x  e.  A  ->  ( ( ph  /\  A. x  e.  A  ps ) 
<->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
51, 4bitr4id 198 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 103    <-> wb 104   F/wnf 1440   E.wex 1472    e. wcel 2128   A.wral 2435
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-ral 2440
This theorem is referenced by:  r19.28mv  3486  raaanlem  3499
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