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Theorem r19.44av 2865
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 2864 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
2 idd 23 . . . 4  |-  ( x  e.  A  ->  ( ps  ->  ps ) )
32rexlimiv 2825 . . 3  |-  ( E. x  e.  A  ps  ->  ps )
43orim2i 506 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( E. x  e.  A  ph  \/  ps ) )
51, 4sylbi 189 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( E. x  e.  A  ph  \/  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    e. wcel 1726   E.wrex 2707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-ral 2711  df-rex 2712
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