Theorem bj-eumo0 32830

Description: Existential uniqueness implies "at most one." Used to be in the main part and deprecated in favor of eumo 2499 and mo2 2479. (Contributed by NM, 8-Jul-1994.) (Revised by BJ, 8-Jun-2019.)

Hypothesis

Ref	Expression
bj-eumo0.1	|- F/ y ph

Assertion

Ref	Expression
bj-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 bj-eumo0

Step	Hyp	Ref	Expression
1		bj-eumo0.1	. . 3 |- F/ y ph
2	1	euf 2478	. 2 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		biimp 205	. . . 4 |- ( ( ph <-> x = y ) -> ( ph -> x = y ) )
4	3	alimi 1739	. . 3 |- ( A. x ( ph <-> x = y ) -> A. x ( ph -> x = y ) )
5	4	eximi 1762	. 2 |- ( E. y A. x ( ph <-> x = y ) -> E. y A. x ( ph -> x = y ) )
6	2, 5	sylbi 207	1 |- ( E! x ph -> E. y A. x ( ph -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   E!weu 2470

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  ax-13 2246

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

This theorem is referenced by: (None)