Theorem euf 2024

Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Hypothesis

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem euf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2022	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		euf.1	. . . . 5 |- ( ph -> A. y ph )
3		ax-17 1519	. . . . 5 |- ( x = z -> A. y x = z )
4	2, 3	hbbi 1541	. . . 4 |- ( ( ph <-> x = z ) -> A. y ( ph <-> x = z ) )
5	4	hbal 1470	. . 3 |- ( A. x ( ph <-> x = z ) -> A. y A. x ( ph <-> x = z ) )
6		ax-17 1519	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z A. x ( ph <-> x = y ) )
7		equequ2 1706	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 231	. . . 4 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1817	. . 3 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	5, 6, 9	cbvexh 1748	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. y A. x ( ph <-> x = y ) )
11	1, 10	bitri 183	1 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   E.wex 1485   E!weu 2019

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-eu 2022

This theorem is referenced by:  eu1  2044  eumo0  2050