Theorem mopick 2334

Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)

Assertion

Ref	Expression
mopick	|- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Proof of Theorem mopick

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2276	. . 3 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )
2		sp 1914	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( ph -> x = y ) )
3		pm3.45 842	. . . . . . 7 |- ( ( ph -> x = y ) -> ( ( ph /\ ps ) -> ( x = y /\ ps ) ) )
4	3	aleximi 1698	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> E. x ( x = y /\ ps ) ) )
5		sb56 2227	. . . . . . 7 |- ( E. x ( x = y /\ ps ) <-> A. x ( x = y -> ps ) )
6		sp 1914	. . . . . . 7 |- ( A. x ( x = y -> ps ) -> ( x = y -> ps ) )
7	5, 6	sylbi 198	. . . . . 6 |- ( E. x ( x = y /\ ps ) -> ( x = y -> ps ) )
8	4, 7	syl6 34	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( x = y -> ps ) ) )
9	2, 8	syl5d 69	. . . 4 |- ( A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
10	9	exlimiv 1770	. . 3 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
11	1, 10	sylbi 198	. 2 |- ( E* x ph -> ( E. x ( ph /\ ps ) -> ( ph -> ps ) ) )
12	11	imp 430	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 370   A.wal 1435   E.wex 1657   E*wmo 2270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-eu 2273  df-mo 2274

This theorem is referenced by:  eupick  2335  mopick2  2339  moexex  2340  morex  3254  imadif  5676  cmetss  22282