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Theorem r19.44av 2709
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 2708 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
2 idd 21 . . . 4  |-  ( x  e.  A  ->  ( ps  ->  ps ) )
32rexlimiv 2674 . . 3  |-  ( E. x  e.  A  ps  ->  ps )
43orim2i 504 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( E. x  e.  A  ph  \/  ps ) )
51, 4sylbi 187 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( E. x  e.  A  ph  \/  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    e. wcel 1696   E.wrex 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-ral 2561  df-rex 2562
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