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Theorem r19.3rzv 3721
Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
Assertion
Ref Expression
r19.3rzv  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Distinct variable groups:    x, A    ph, x

Proof of Theorem r19.3rzv
StepHypRef Expression
1 n0 3637 . . 3  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
2 biimt 326 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
31, 2sylbi 188 . 2  |-  ( A  =/=  (/)  ->  ( ph  <->  ( E. x  x  e.  A  ->  ph ) ) )
4 df-ral 2710 . . 3  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
5 19.23v 1914 . . 3  |-  ( A. x ( x  e.  A  ->  ph )  <->  ( E. x  x  e.  A  ->  ph ) )
64, 5bitri 241 . 2  |-  ( A. x  e.  A  ph  <->  ( E. x  x  e.  A  ->  ph ) )
73, 6syl6bbr 255 1  |-  ( A  =/=  (/)  ->  ( ph  <->  A. x  e.  A  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    e. wcel 1725    =/= wne 2599   A.wral 2705   (/)c0 3628
This theorem is referenced by:  r19.9rzv  3722  r19.28zv  3723  r19.37zv  3724  r19.27zv  3727  iinconst  4102  cnvpo  5410  coe1mul2lem1  16660  neipeltop  17193  utop3cls  18281  rencldnfi  26882
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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