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Theorem r19.37zv 3667
Description: Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)
Assertion
Ref Expression
r19.37zv  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  E. x  e.  A  ps ) ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem r19.37zv
StepHypRef Expression
1 r19.3rzv 3664 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
21imbi1d 309 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  ->  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps )
) )
3 r19.35 2798 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
42, 3syl6rbbr 256 1  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  E. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    =/= wne 2550   A.wral 2649   E.wrex 2650   (/)c0 3571
This theorem is referenced by:  ishlat3N  29469  hlsupr2  29501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-dif 3266  df-nul 3572
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