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Theorem r19.43 2666
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 2658 . 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 2539 . 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 2524 . . . 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 2516   E.wrex 2517
This theorem is referenced by:  r19.44av  2667  r19.45av  2668  r19.45zv  3493  iunun  3923  wemapso2lem  7198  pythagtriplem2  12797  pythagtrip  12814  dcubic  20069  erdszelem11  23069  soseq  23588  axcontlem4  23935  seglelin  24079  diophun  26185  rexzrexnn0  26217
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 2520  df-rex 2521
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