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Theorem r19.27z 3726
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, 26-Oct-2010.)
Hypothesis
Ref Expression
r19.27z.1  |-  F/ x ps
Assertion
Ref Expression
r19.27z  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem r19.27z
StepHypRef Expression
1 r19.27z.1 . . . 4  |-  F/ x ps
21r19.3rz 3719 . . 3  |-  ( A  =/=  (/)  ->  ( ps  <->  A. x  e.  A  ps ) )
32anbi2d 685 . 2  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
/\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
4 r19.26 2838 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4syl6rbbr 256 1  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    =/= wne 2599   A.wral 2705   (/)c0 3628
This theorem is referenced by:  raaan  3735  raaan2  27929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-v 2958  df-dif 3323  df-nul 3629
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