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

Step	Hyp	Ref	Expression
1		euor.1	. . . 4 |- ( ph -> A. x ph )
2	1	hbn 1589	. . 3 |- ( -. ph -> A. x -. ph )
3		biorf 698	. . 3 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
4	2, 3	eubidh 1954	. 2 |- ( -. ph -> ( E! x ps <-> E! x ( ph \/ ps ) ) )
5	4	biimpa 290	1 |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )

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