MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  19.30 Structured version   Unicode version

Theorem 19.30 1659
Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
19.30  |-  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) )

Proof of Theorem 19.30
StepHypRef Expression
1 exnal 1618 . . 3  |-  ( E. x  -.  ph  <->  -.  A. x ph )
2 pm2.53 373 . . . 4  |-  ( (
ph  \/  ps )  ->  ( -.  ph  ->  ps ) )
32aleximi 1622 . . 3  |-  ( A. x ( ph  \/  ps )  ->  ( E. x  -.  ph  ->  E. x ps ) )
41, 3syl5bir 218 . 2  |-  ( A. x ( ph  \/  ps )  ->  ( -. 
A. x ph  ->  E. x ps ) )
54orrd 378 1  |-  ( A. x ( ph  \/  ps )  ->  ( A. x ph  \/  E. x ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368   A.wal 1367   E.wex 1586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602
This theorem depends on definitions:  df-bi 185  df-or 370  df-ex 1587
This theorem is referenced by:  19.33b  1663
  Copyright terms: Public domain W3C validator