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Theorem r19.37zv 3550
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 3547 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
21imbi1d 308 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  ->  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps )
) )
3 r19.35 2687 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
42, 3syl6rbbr 255 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 176    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455
This theorem is referenced by:  ishlat3N  29544  hlsupr2  29576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456
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