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Theorem 19.43 1604
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
Assertion
Ref Expression
19.43  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )

Proof of Theorem 19.43
StepHypRef Expression
1 df-or 361 . . . 4  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
21exbii 1580 . . 3  |-  ( E. x ( ph  \/  ps )  <->  E. x ( -. 
ph  ->  ps ) )
3 19.35 1599 . . 3  |-  ( E. x ( -.  ph  ->  ps )  <->  ( A. x  -.  ph  ->  E. x ps ) )
4 alnex 1569 . . . 4  |-  ( A. x  -.  ph  <->  -.  E. x ph )
54imbi1i 317 . . 3  |-  ( ( A. x  -.  ph  ->  E. x ps )  <->  ( -.  E. x ph  ->  E. x ps )
)
62, 3, 53bitri 264 . 2  |-  ( E. x ( ph  \/  ps )  <->  ( -.  E. x ph  ->  E. x ps ) )
7 df-or 361 . 2  |-  ( ( E. x ph  \/  E. x ps )  <->  ( -.  E. x ph  ->  E. x ps ) )
86, 7bitr4i 245 1  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359   A.wal 1532   E.wex 1537
This theorem is referenced by:  19.44  1796  19.45  1797  19.34  1798  rexun  3297  unipr  3782  uniun  3787  unopab  4035  zfpair  4150  dmun  4838  coundi  5126  coundir  5127  kmlem16  7724  vdwapun  12948  pm10.42  26891
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-ex 1538
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