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Theorem r19.37zv 3725
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 3722 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
21imbi1d 310 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  ->  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps )
) )
3 r19.35 2856 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
42, 3syl6rbbr 257 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 178    =/= wne 2600   A.wral 2706   E.wrex 2707   (/)c0 3629
This theorem is referenced by:  ishlat3N  30153  hlsupr2  30185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-v 2959  df-dif 3324  df-nul 3630
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