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Theorem r19.27z 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, 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 groups:    ph( x)    ps( x)

Proof of Theorem r19.27z
StepHypRef Expression
1 r19.27z.1 . . . 4  |-  F/ x ps
21r19.3rz 3548 . . 3  |-  ( A  =/=  (/)  ->  ( ps  <->  A. x  e.  A  ps ) )
32anbi2d 686 . 2  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
/\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) ) )
4 r19.26 2678 . 2  |-  ( A. x  e.  A  ( ph  /\  ps )  <->  ( A. x  e.  A  ph  /\  A. x  e.  A  ps ) )
53, 4syl6rbbr 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   F/wnf 1533    =/= wne 2449   A.wral 2546   (/)c0 3458
This theorem is referenced by:  raaan  3564  raaan2  27334
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-v 2793  df-dif 3158  df-nul 3459
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