Theorem euor 2040

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 1642	. . 3 |- ( -. ph -> A. x -. ph )
3		biorf 734	. . 3 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
4	2, 3	eubidh 2020	. 2 |- ( -. ph -> ( E! x ps <-> E! x ( ph \/ ps ) ) )
5	4	biimpa 294	1 |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   A.wal 1341   E!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-eu 2017

This theorem is referenced by:  euorv  2041