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Theorem r19.35 2700
Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
r19.35  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )

Proof of Theorem r19.35
StepHypRef Expression
1 r19.26 2688 . . . 4  |-  ( A. x  e.  A  ( ph  /\  -.  ps )  <->  ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ps ) )
2 annim 414 . . . . 5  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) )
32ralbii 2580 . . . 4  |-  ( A. x  e.  A  ( ph  /\  -.  ps )  <->  A. x  e.  A  -.  ( ph  ->  ps )
)
4 df-an 360 . . . 4  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ps )  <->  -.  ( A. x  e.  A  ph 
->  -.  A. x  e.  A  -.  ps )
)
51, 3, 43bitr3i 266 . . 3  |-  ( A. x  e.  A  -.  ( ph  ->  ps )  <->  -.  ( A. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ps ) )
65con2bii 322 . 2  |-  ( ( A. x  e.  A  ph 
->  -.  A. x  e.  A  -.  ps )  <->  -. 
A. x  e.  A  -.  ( ph  ->  ps ) )
7 dfrex2 2569 . . 3  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
87imbi2i 303 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ps ) )
9 dfrex2 2569 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  -.  A. x  e.  A  -.  ( ph  ->  ps ) )
106, 8, 93bitr4ri 269 1  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wral 2556   E.wrex 2557
This theorem is referenced by:  r19.36av  2701  r19.37  2702  r19.43  2708  r19.37zv  3563  r19.36zv  3567  iinexg  4187  bndndx  9980  nmobndseqi  21373  nmobndseqiOLD  21374  intopcoaconlem3b  25641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-ral 2561  df-rex 2562
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