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Theorem r19.45mv 3424
Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.45mv  |-  ( E. x  x  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  E. x  e.  A  ps ) ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem r19.45mv
StepHypRef Expression
1 r19.9rmv 3422 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  E. x  e.  A  ph ) )
21orbi1d 763 . 2  |-  ( E. x  x  e.  A  ->  ( ( ph  \/  E. x  e.  A  ps ) 
<->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) ) )
3 r19.43 2564 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
42, 3syl6rbbr 198 1  |-  ( E. x  x  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  E. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 680   E.wex 1451    e. wcel 1463   E.wrex 2392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-clel 2111  df-rex 2397
This theorem is referenced by:  ltexprlemloc  7379
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