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Theorem r19.36zv 3556
Description: Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
r19.36zv  |-  ( A  =/=  (/)  ->  ( E. 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 r19.36zv
StepHypRef Expression
1 r19.9rzv 3550 . . 3  |-  ( A  =/=  (/)  ->  ( ps  <->  E. x  e.  A  ps ) )
21imbi2d 307 . 2  |-  ( A  =/=  (/)  ->  ( ( A. x  e.  A  ph 
->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) ) )
3 r19.35 2689 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
42, 3syl6rbbr 255 1  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    =/= wne 2448   A.wral 2545   E.wrex 2546   (/)c0 3457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-v 2792  df-dif 3157  df-nul 3458
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