Theorem moaneu 2533

Description: Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)

Assertion

Ref	Expression
moaneu	|- E* x ( ph /\ E! x ph )

Proof of Theorem moaneu

Step	Hyp	Ref	Expression
1		moanmo 2532	. 2 |- E* x ( ph /\ E* x ph )
2		eumo 2499	. . . 4 |- ( E! x ph -> E* x ph )
3	2	anim2i 593	. . 3 |- ( ( ph /\ E! x ph ) -> ( ph /\ E* x ph ) )
4	3	moimi 2520	. 2 |- ( E* x ( ph /\ E* x ph ) -> E* x ( ph /\ E! x ph ) )
5	1, 4	ax-mp 5	1 |- E* x ( ph /\ E! x ph )

Syntax hints:   /\ wa 384   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)