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Theorem r19.30 2698
Description: Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
Assertion
Ref Expression
r19.30  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )

Proof of Theorem r19.30
StepHypRef Expression
1 ralim 2627 . 2  |-  ( A. x  e.  A  ( -.  ps  ->  ph )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
2 orcom 376 . . . 4  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
3 df-or 359 . . . 4  |-  ( ( ps  \/  ph )  <->  ( -.  ps  ->  ph )
)
42, 3bitri 240 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ps  ->  ph )
)
54ralbii 2580 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ps  ->  ph ) )
6 orcom 376 . . 3  |-  ( ( A. x  e.  A  ph  \/  -.  A. x  e.  A  -.  ps )  <->  ( -.  A. x  e.  A  -.  ps  \/  A. x  e.  A  ph ) )
7 dfrex2 2569 . . . 4  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
87orbi2i 505 . . 3  |-  ( ( A. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  \/  -.  A. x  e.  A  -.  ps ) )
9 imor 401 . . 3  |-  ( ( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph )  <->  ( -.  A. x  e.  A  -.  ps  \/  A. x  e.  A  ph ) )
106, 8, 93bitr4i 268 . 2  |-  ( ( A. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
111, 5, 103imtr4i 257 1  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357   A.wral 2556   E.wrex 2557
This theorem is referenced by:  esumcvg  23469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-ral 2561  df-rex 2562
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