Theorem eumo0 2057

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 2031	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		biimp 118	. . . 4 |- ( ( ph <-> x = y ) -> ( ph -> x = y ) )
4	3	alimi 1455	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x ( ph -> x = y ) )
5	4	eximi 1600	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y A. x ( ph -> x = y ) )
6	2, 5	sylbi 121	1 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   E.wex 1492   E!weu 2026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-eu 2029

This theorem is referenced by:  eu2  2070  eu3h  2071