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Theorem r19.43 2635
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 2461 . . . 4  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x ( x  e.  A  /\  ( ph  \/  ps ) ) )
2 andi 818 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  \/  ps )
)  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
32exbii 1605 . . . 4  |-  ( E. x ( x  e.  A  /\  ( ph  \/  ps ) )  <->  E. x
( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) )
41, 3bitri 184 . . 3  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
5 19.43 1628 . . 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 184 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x ( x  e.  A  /\  ph )  \/  E. x
( x  e.  A  /\  ps ) ) )
7 df-rex 2461 . . 3  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2461 . . 3  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
97, 8orbi12i 764 . 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 187 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 104    <-> wb 105    \/ wo 708   E.wex 1492    e. wcel 2148   E.wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-rex 2461
This theorem is referenced by:  r19.44av  2636  r19.45av  2637  r19.45mv  3516  r19.44mv  3517  iunun  3962  ltexprlemloc  7594  pythagtriplem2  12246  pythagtrip  12263
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