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

Theorem r19.43 3017
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 3008 . 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 2965 . 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 2910 . . . 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 2814   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-ral 2819  df-rex 2820
This theorem is referenced by:  r19.44av  3018  r19.45av  3019  r19.45zv  3925  iunun  4406  wemapsolem  7971  pythagtriplem2  14193  pythagtrip  14210  dcubic  22902  legtrid  23702  axcontlem4  23943  erdszelem11  28282  soseq  28908  seglelin  29340  diophun  30309  rexzrexnn0  30339  usgvincvad  31873  usgvincvadALT  31876  ldepslinc  32191
  Copyright terms: Public domain W3C validator