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Theorem r19.27zv 3555
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 3549 . . 3  |-  ( A  =/=  (/)  ->  ( ps  <->  A. x  e.  A  ps ) )
21anbi2d 686 . 2  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
/\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
3 r19.26 2677 . 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 )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    =/= wne 2448   A.wral 2545   (/)c0 3457
This theorem is referenced by:  raaanv  3564  txflf  17696  dfso3  23479  dibglbN  30624
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 1868  ax-ext 2266
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 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-v 2792  df-dif 3157  df-nul 3458
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