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Theorem r19.44mv 3352
Description: Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.44mv  |-  ( E. y  y  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  ps ) ) )
Distinct variable groups:    x, A    y, A    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)

Proof of Theorem r19.44mv
StepHypRef Expression
1 r19.9rmv 3349 . . 3  |-  ( E. y  y  e.  A  ->  ( ps  <->  E. x  e.  A  ps )
)
21orbi2d 737 . 2  |-  ( E. y  y  e.  A  ->  ( ( E. x  e.  A  ph  \/  ps ) 
<->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) ) )
3 r19.43 2517 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
42, 3syl6rbbr 197 1  |-  ( E. y  y  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    \/ wo 662   E.wex 1422    e. wcel 1434   E.wrex 2354
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079  df-rex 2359
This theorem is referenced by:  frecabcl  6068
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