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Theorem r19.28zv 3550
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 3548 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
21anbi1d 687 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  /\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
3 r19.26 2676 . 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 2447   A.wral 2544   (/)c0 3456
This theorem is referenced by:  raaanv  3563  iinrab  3965  iindif2  3972  iinin2  3973  reusv2lem5  4538  reusv7OLD  4545  xpiindi  4820  fint  5385  ixpiin  6837  neips  16844  txflf  17695  dfpo2  23515  diaglbN  30512  dihglbcpreN  30757
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-v 2791  df-dif 3156  df-nul 3457
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