Theorem euf 2318

Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.)

Hypothesis

Ref	Expression
euf.1	|- F/ 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 2314	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		euf.1	. . . . 5 |- F/ y ph
3		nfv 1772	. . . . 5 |- F/ y x = z
4	2, 3	nfbi 2028	. . . 4 |- F/ y ( ph <-> x = z )
5	4	nfal 2041	. . 3 |- F/ y A. x ( ph <-> x = z )
6		nfv 1772	. . 3 |- F/ z A. x ( ph <-> x = y )
7		equequ2 1879	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 324	. . . 4 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1778	. . 3 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	5, 6, 9	cbvex 2126	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. y A. x ( ph <-> x = y ) )
11	1, 10	bitri 257	1 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

Syntax hints:   <-> wb 189   A.wal 1453   E.wex 1674   F/wnf 1678   E!weu 2310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-eu 2314

This theorem is referenced by:  eu1  2350  bj-eumo0  31488