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Theorem r19.45mv 3375
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 3373 . . 3  |-  ( E. x  x  e.  A  ->  ( ph  <->  E. x  e.  A  ph ) )
21orbi1d 740 . 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 2525 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
42, 3syl6rbbr 197 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 103    \/ wo 664   E.wex 1426    e. wcel 1438   E.wrex 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-rex 2365
This theorem is referenced by:  ltexprlemloc  7164
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