Theorem euf 2004

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 2002	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		euf.1	. . . . 5 |- ( ph -> A. y ph )
3		ax-17 1506	. . . . 5 |- ( x = z -> A. y x = z )
4	2, 3	hbbi 1527	. . . 4 |- ( ( ph <-> x = z ) -> A. y ( ph <-> x = z ) )
5	4	hbal 1453	. . 3 |- ( A. x ( ph <-> x = z ) -> A. y A. x ( ph <-> x = z ) )
6		ax-17 1506	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z A. x ( ph <-> x = y ) )
7		equequ2 1689	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 231	. . . 4 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1796	. . 3 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	5, 6, 9	cbvexh 1728	. 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 1329   E.wex 1468   E!weu 1999

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-eu 2002

This theorem is referenced by:  eu1  2024  eumo0  2030