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Theorem r19.35 2689
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 2677 . . . 4  |-  ( A. x  e.  A  ( ph  /\  -.  ps )  <->  ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ps ) )
2 annim 416 . . . . 5  |-  ( (
ph  /\  -.  ps )  <->  -.  ( ph  ->  ps ) )
32ralbii 2569 . . . 4  |-  ( A. x  e.  A  ( ph  /\  -.  ps )  <->  A. x  e.  A  -.  ( ph  ->  ps )
)
4 df-an 362 . . . 4  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  A  -.  ps )  <->  -.  ( A. x  e.  A  ph 
->  -.  A. x  e.  A  -.  ps )
)
51, 3, 43bitr3i 268 . . 3  |-  ( A. x  e.  A  -.  ( ph  ->  ps )  <->  -.  ( A. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ps ) )
65con2bii 324 . 2  |-  ( ( A. x  e.  A  ph 
->  -.  A. x  e.  A  -.  ps )  <->  -. 
A. x  e.  A  -.  ( ph  ->  ps ) )
7 dfrex2 2558 . . 3  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
87imbi2i 305 . 2  |-  ( ( A. x  e.  A  ph 
->  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ps ) )
9 dfrex2 2558 . 2  |-  ( E. x  e.  A  (
ph  ->  ps )  <->  -.  A. x  e.  A  -.  ( ph  ->  ps ) )
106, 8, 93bitr4ri 271 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 5    -> wi 6    <-> wb 178    /\ wa 360   A.wral 2545   E.wrex 2546
This theorem is referenced by:  r19.36av  2690  r19.37  2691  r19.43  2697  r19.37zv  3552  r19.36zv  3556  iinexg  4175  bndndx  9960  nmobndseqi  21350  nmobndseqiOLD  21351  intopcoaconlem3b  24938
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-11 1716
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-ral 2550  df-rex 2551
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