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Theorem euor 1974
Description: Introduce a disjunct into a unique existential quantifier. (Contributed by NM, 21-Oct-2005.)
Hypothesis
Ref Expression
euor.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
euor  |-  ( ( -.  ph  /\  E! x ps )  ->  E! x
( ph  \/  ps ) )

Proof of Theorem euor
StepHypRef Expression
1 euor.1 . . . 4  |-  ( ph  ->  A. x ph )
21hbn 1589 . . 3  |-  ( -. 
ph  ->  A. x  -.  ph )
3 biorf 698 . . 3  |-  ( -. 
ph  ->  ( ps  <->  ( ph  \/  ps ) ) )
42, 3eubidh 1954 . 2  |-  ( -. 
ph  ->  ( E! x ps 
<->  E! x ( ph  \/  ps ) ) )
54biimpa 290 1  |-  ( ( -.  ph  /\  E! x ps )  ->  E! x
( ph  \/  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664   A.wal 1287   E!weu 1948
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-eu 1951
This theorem is referenced by:  euorv  1975
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