Theorem euor 2052

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 1654	. . 3 |- ( -. ph -> A. x -. ph )
3		biorf 744	. . 3 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
4	2, 3	eubidh 2032	. 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 708   A.wal 1351   E!weu 2026

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-eu 2029

This theorem is referenced by:  euorv  2053