Theorem eumo0 2045

Description: Existential uniqueness implies "at most one". (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eumo0.1	|- ( ph -> A. y ph )

Assertion

Ref	Expression
eumo0	|- ( E! x ph -> E. y A. x ( ph -> x = y ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 |- ( ph -> A. y ph )
2	1	euf 2019	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		biimp 117	. . . 4 |- ( ( ph <-> x = y ) -> ( ph -> x = y ) )
4	3	alimi 1443	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x ( ph -> x = y ) )
5	4	eximi 1588	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y A. x ( ph -> x = y ) )
6	2, 5	sylbi 120	1 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   E.wex 1480   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-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-eu 2017

This theorem is referenced by:  eu2  2058  eu3h  2059