Theorem eumo0 1931

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 1905	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		bi1 111	. . . 4 |- ( ( ph <-> x = y ) -> ( ph -> x = y ) )
4	3	alimi 1344	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x ( ph -> x = y ) )
5	4	eximi 1491	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y A. x ( ph -> x = y ) )
6	2, 5	sylbi 114	1 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   E.wex 1381   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-eu 1903

This theorem is referenced by:  eu2  1944  eu3h  1945