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Theorem r19.43 2564
Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
Assertion
Ref Expression
r19.43  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )

Proof of Theorem r19.43
StepHypRef Expression
1 df-rex 2397 . . . 4  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x ( x  e.  A  /\  ( ph  \/  ps ) ) )
2 andi 790 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  \/  ps )
)  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
32exbii 1567 . . . 4  |-  ( E. x ( x  e.  A  /\  ( ph  \/  ps ) )  <->  E. x
( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) )
41, 3bitri 183 . . 3  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
5 19.43 1590 . . 3  |-  ( E. x ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
)  <->  ( E. x
( x  e.  A  /\  ph )  \/  E. x ( x  e.  A  /\  ps )
) )
64, 5bitri 183 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x ( x  e.  A  /\  ph )  \/  E. x
( x  e.  A  /\  ps ) ) )
7 df-rex 2397 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2397 . . 3  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
97, 8orbi12i 736 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( E. x ( x  e.  A  /\  ph )  \/  E. x ( x  e.  A  /\  ps ) ) )
106, 9bitr4i 186 1  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 680   E.wex 1451    e. wcel 1463   E.wrex 2392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-rex 2397
This theorem is referenced by:  r19.44av  2565  r19.45av  2566  r19.45mv  3424  r19.44mv  3425  iunun  3859  ltexprlemloc  7379
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