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Theorem r19.28zv 3510
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, 19-Aug-2004.)
Assertion
Ref Expression
r19.28zv  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( ph  /\ 
A. x  e.  A  ps ) ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem r19.28zv
StepHypRef Expression
1 r19.3rzv 3508 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
21anbi1d 688 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  /\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
3 r19.26 2648 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
42, 3syl6rbbr 257 1  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( ph  /\ 
A. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    =/= wne 2419   A.wral 2516   (/)c0 3416
This theorem is referenced by:  raaanv  3523  iinrab  3924  iindif2  3931  iinin2  3932  reusv2lem5  4497  reusv7OLD  4504  xpiindi  4795  fint  5344  ixpiin  6796  neips  16798  txflf  17649  dfpo2  23469  diaglbN  30396  dihglbcpreN  30641
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-v 2759  df-dif 3116  df-nul 3417
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