Theorem euor 2071

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 1668	. . 3 |- ( -. ph -> A. x -. ph )
3		biorf 745	. . 3 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
4	2, 3	eubidh 2051	. 2 |- ( -. ph -> ( E! x ps <-> E! x ( ph \/ ps ) ) )
5	4	biimpa 296	1 |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709   A.wal 1362   E!weu 2045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-eu 2048

This theorem is referenced by:  euorv  2072