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Theorem r19.43 2670
Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
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 r19.35 2662 . 2  |-  ( E. x  e.  A  ( -.  ph  ->  ps )  <->  ( A. x  e.  A  -.  ph  ->  E. x  e.  A  ps )
)
2 df-or 361 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32rexbii 2543 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x  e.  A  ( -.  ph  ->  ps )
)
4 df-or 361 . . 3  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( -.  E. x  e.  A  ph  ->  E. x  e.  A  ps ) )
5 ralnex 2528 . . . 4  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
65imbi1i 317 . . 3  |-  ( ( A. x  e.  A  -.  ph  ->  E. x  e.  A  ps )  <->  ( -.  E. x  e.  A  ph  ->  E. x  e.  A  ps )
)
74, 6bitr4i 245 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  -.  ph 
->  E. x  e.  A  ps ) )
81, 3, 73bitr4i 270 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:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359   A.wral 2518   E.wrex 2519
This theorem is referenced by:  r19.44av  2671  r19.45av  2672  r19.45zv  3526  iunun  3956  wemapso2lem  7233  pythagtriplem2  12832  pythagtrip  12849  dcubic  20104  erdszelem11  23104  soseq  23623  axcontlem4  23970  seglelin  24114  diophun  26220  rexzrexnn0  26252
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-ral 2523  df-rex 2524
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