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Theorem r19.28z 3712
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
Hypothesis
Ref Expression
r19.3rz.1  |-  F/ x ph
Assertion
Ref Expression
r19.28z  |-  ( 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 r19.28z
StepHypRef Expression
1 r19.3rz.1 . . . 4  |-  F/ x ph
21r19.3rz 3711 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
32anbi1d 686 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  /\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
4 r19.26 2830 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4syl6rbbr 256 1  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( ph  /\ 
A. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    =/= wne 2598   A.wral 2697   (/)c0 3620
This theorem is referenced by:  raaan  3727  raaan2  27867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-nul 3621
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