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Theorem r19.28zv 3929
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 3927 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
21anbi1d 704 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  /\  A. x  e.  A  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
3 r19.26 2994 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
42, 3syl6rbbr 264 1  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( ph  /\ 
A. x  e.  A  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    =/= wne 2662   A.wral 2817   (/)c0 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-v 3120  df-dif 3484  df-nul 3791
This theorem is referenced by:  raaanv  3942  raltpd  4156  iinrab  4393  iindif2  4400  iinin2  4401  reusv2lem5  4658  reusv7OLD  4665  xpiindi  5144  fint  5770  ixpiin  7507  neips  19482  txflf  20375  dfpo2  29111  diaglbN  36253  dihglbcpreN  36498
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