Theorem euf 2228

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 2222	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		euf.1	. . . . 5 |- F/ y ph
3		nfv 1715	. . . . 5 |- F/ y x = z
4	2, 3	nfbi 1942	. . . 4 |- F/ y ( ph <-> x = z )
5	4	nfal 1955	. . 3 |- F/ y A. x ( ph <-> x = z )
6		nfv 1715	. . 3 |- F/ z A. x ( ph <-> x = y )
7		equequ2 1807	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
8	7	bibi2d 316	. . . 4 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
9	8	albidv 1721	. . 3 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
10	5, 6, 9	cbvex 2029	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. y A. x ( ph <-> x = y ) )
11	1, 10	bitri 249	1 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

Syntax hints:   <-> wb 184   A.wal 1397   E.wex 1620   F/wnf 1624   E!weu 2218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-eu 2222

This theorem is referenced by:  eu1  2262  bj-eumo0  34761