MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.44av Structured version   Unicode version

Theorem r19.44av 3018
Description: One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when  A is empty. (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.44av  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( E. x  e.  A  ph  \/  ps ) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem r19.44av
StepHypRef Expression
1 r19.43 3017 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
2 idd 24 . . . 4  |-  ( x  e.  A  ->  ( ps  ->  ps ) )
32rexlimiv 2949 . . 3  |-  ( E. x  e.  A  ps  ->  ps )
43orim2i 518 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( E. x  e.  A  ph  \/  ps ) )
51, 4sylbi 195 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( E. x  e.  A  ph  \/  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    e. wcel 1767   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  ax-5 1680
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: (None)
  Copyright terms: Public domain W3C validator