Theorem exmoeu 2502

Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)

Assertion

Ref	Expression
exmoeu	|- ( E. x ph <-> ( E* x ph -> E! x ph ) )

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		df-mo 2475	. . . 4 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
2	1	biimpi 206	. . 3 |- ( E* x ph -> ( E. x ph -> E! x ph ) )
3	2	com12 32	. 2 |- ( E. x ph -> ( E* x ph -> E! x ph ) )
4		exmo 2495	. . . . 5 |- ( E. x ph \/ E* x ph )
5	4	ori 390	. . . 4 |- ( -. E. x ph -> E* x ph )
6	5	con1i 144	. . 3 |- ( -. E* x ph -> E. x ph )
7		euex 2494	. . 3 |- ( E! x ph -> E. x ph )
8	6, 7	ja 173	. 2 |- ( ( E* x ph -> E! x ph ) -> E. x ph )
9	3, 8	impbii 199	1 |- ( E. x ph <-> ( E* x ph -> E! x ph ) )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)