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Theorem r19.45av 2857
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 2855 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
2 idd 22 . . . 4  |-  ( x  e.  A  ->  ( ph  ->  ph ) )
32rexlimiv 2816 . . 3  |-  ( E. x  e.  A  ph  ->  ph )
43orim1i 504 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( ph  \/  E. x  e.  A  ps )
)
51, 4sylbi 188 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    e. wcel 1725   E.wrex 2698
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-ral 2702  df-rex 2703
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