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Theorem r19.45av 2617
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 2615 . 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 2568 . . 3  |-  ( E. x  e.  A  ph  ->  ph )
43orim1i 750 . 2  |-  ( ( E. x  e.  A  ph  \/  E. x  e.  A  ps )  -> 
( ph  \/  E. x  e.  A  ps )
)
51, 4sylbi 120 1  |-  ( E. x  e.  A  (
ph  \/  ps )  ->  ( ph  \/  E. x  e.  A  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 698    e. wcel 2128   E.wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440  df-rex 2441
This theorem is referenced by: (None)
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