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Theorem r19.43 2525
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 2365 . . . 4  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x ( x  e.  A  /\  ( ph  \/  ps ) ) )
2 andi 767 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  \/  ps )
)  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
32exbii 1541 . . . 4  |-  ( E. x ( x  e.  A  /\  ( ph  \/  ps ) )  <->  E. x
( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) )
41, 3bitri 182 . . 3  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
5 19.43 1564 . . 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 182 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x ( x  e.  A  /\  ph )  \/  E. x
( x  e.  A  /\  ps ) ) )
7 df-rex 2365 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2365 . . 3  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
97, 8orbi12i 716 . 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 185 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 102    <-> wb 103    \/ wo 664   E.wex 1426    e. wcel 1438   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-rex 2365
This theorem is referenced by:  r19.44av  2526  r19.45av  2527  r19.45mv  3371  r19.44mv  3372  iunun  3803  ltexprlemloc  7145
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