MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  r19.45av Unicode version

Theorem r19.45av 2697
Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when  A is empty.) (Contributed by NM, 2-Apr-2004.)
Assertion
Ref Expression
r19.45av  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem r19.45av
StepHypRef Expression
1 r19.43 2695 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
2 idd 21 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
32rexlimiv 2661 . . 3  |-  ( E. x  e.  A  ph  ->  ph )
43orim1i 503 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( ph  \/  E. x  e.  A  ps )
)
51, 4sylbi 187 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    e. wcel 1684   E.wrex 2544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-ral 2548  df-rex 2549
  Copyright terms: Public domain W3C validator