MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.30 Structured version   Unicode version

Theorem r19.30 2970
Description: Restricted quantifier version of 19.30 1738. (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 2811 . 2  |-  ( A. x  e.  A  ( -.  ps  ->  ph )  -> 
( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
2 orcom 388 . . . 4  |-  ( (
ph  \/  ps )  <->  ( ps  \/  ph )
)
3 df-or 371 . . . 4  |-  ( ( ps  \/  ph )  <->  ( -.  ps  ->  ph )
)
42, 3bitri 252 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ps  ->  ph )
)
54ralbii 2853 . 2  |-  ( A. x  e.  A  ( ph  \/  ps )  <->  A. x  e.  A  ( -.  ps  ->  ph ) )
6 orcom 388 . . 3  |-  ( ( A. x  e.  A  ph  \/  -.  A. x  e.  A  -.  ps )  <->  ( -.  A. x  e.  A  -.  ps  \/  A. x  e.  A  ph ) )
7 dfrex2 2873 . . . 4  |-  ( E. x  e.  A  ps  <->  -. 
A. x  e.  A  -.  ps )
87orbi2i 521 . . 3  |-  ( ( A. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  ph  \/  -.  A. x  e.  A  -.  ps ) )
9 imor 413 . . 3  |-  ( ( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph )  <->  ( -.  A. x  e.  A  -.  ps  \/  A. x  e.  A  ph ) )
106, 8, 93bitr4i 280 . 2  |-  ( ( A. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  -.  ps  ->  A. x  e.  A  ph ) )
111, 5, 103imtr4i 269 1  |-  ( A. x  e.  A  ( ph  \/  ps )  -> 
( A. x  e.  A  ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369   A.wral 2771   E.wrex 2772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-ral 2776  df-rex 2777
This theorem is referenced by:  disjunsn  28206  esumcvg  28915
  Copyright terms: Public domain W3C validator