MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.43 Structured version   Unicode version

Theorem r19.43 2881
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 2872 . 2  |-  ( E. x  e.  A  ( -.  ph  ->  ps )  <->  ( A. x  e.  A  -.  ph  ->  E. x  e.  A  ps )
)
2 df-or 370 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
32rexbii 2745 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  E. x  e.  A  ( -.  ph  ->  ps )
)
4 df-or 370 . . 3  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( -.  E. x  e.  A  ph  ->  E. x  e.  A  ps ) )
5 ralnex 2730 . . . 4  |-  ( A. x  e.  A  -.  ph  <->  -. 
E. x  e.  A  ph )
65imbi1i 325 . . 3  |-  ( ( A. x  e.  A  -.  ph  ->  E. x  e.  A  ps )  <->  ( -.  E. x  e.  A  ph  ->  E. x  e.  A  ps )
)
74, 6bitr4i 252 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  <->  ( A. x  e.  A  -.  ph 
->  E. x  e.  A  ps ) )
81, 3, 73bitr4i 277 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 184    \/ wo 368   A.wral 2720   E.wrex 2721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-ral 2725  df-rex 2726
This theorem is referenced by:  r19.44av  2882  r19.45av  2883  r19.45zv  3782  iunun  4256  wemapsolem  7769  pythagtriplem2  13889  pythagtrip  13906  dcubic  22246  legtrid  23027  axcontlem4  23218  erdszelem11  27094  soseq  27720  seglelin  28152  diophun  29117  rexzrexnn0  29147  ldepslinc  31056
  Copyright terms: Public domain W3C validator