MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.27zv Structured version   Unicode version

Theorem r19.27zv 3800
Description: Restricted quantifier version of Theorem 19.27 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
r19.27zv  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem r19.27zv
StepHypRef Expression
1 r19.3rzv 3794 . . 3  |-  ( A  =/=  (/)  ->  ( ps  <->  A. x  e.  A  ps ) )
21anbi2d 703 . 2  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
/\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
3 r19.26 2870 . 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 )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    =/= wne 2620   A.wral 2736   (/)c0 3658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-v 2995  df-dif 3352  df-nul 3659
This theorem is referenced by:  raaanv  3809  txflf  19601  dfso3  27398  dibglbN  34907
  Copyright terms: Public domain W3C validator