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Theorem r19.43 2991
Description: Restricted quantifier version of 19.43 1740. (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 2982 . 2  |-  ( E. x  e.  A  ( -.  ph  ->  ps )  <->  ( A. x  e.  A  -.  ph  ->  E. x  e.  A  ps )
)
2 df-or 371 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32rexbii 2934 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x  e.  A  ( -.  ph  ->  ps )
)
4 df-or 371 . . 3  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( -.  E. x  e.  A  ph  ->  E. x  e.  A  ps ) )
5 ralnex 2878 . . . 4  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
65imbi1i 326 . . 3  |-  ( ( A. x  e.  A  -.  ph  ->  E. x  e.  A  ps )  <->  ( -.  E. x  e.  A  ph  ->  E. x  e.  A  ps )
)
74, 6bitr4i 255 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  -.  ph 
->  E. x  e.  A  ps ) )
81, 3, 73bitr4i 280 1  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369   A.wral 2782   E.wrex 2783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1660  df-ral 2787  df-rex 2788
This theorem is referenced by:  r19.44v  2992  r19.45v  2993  r19.45zv  3900  r19.44zv  3901  iunun  4386  wemapsolem  8065  pythagtriplem2  14730  pythagtrip  14747  dcubic  23637  legtrid  24496  axcontlem4  24843  erdszelem11  29712  soseq  30279  seglelin  30668  diophun  35324  rexzrexnn0  35355  usgvincvad  38473  usgvincvadALT  38476  ldepslinc  39061
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