Theorem euf 2260

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	|- 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 2258	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		euf.1	. . . . 5 |- F/ y ph
3		nfv 1626	. . . . 5 |- F/ y x = z
4	2, 3	nfbi 1852	. . . 4 |- F/ y ( ph <-> x = z )
5	4	nfal 1860	. . 3 |- F/ y A. x ( ph <-> x = z )
6		nfv 1626	. . . . 5 |- F/ z ph
7		nfv 1626	. . . . 5 |- F/ z x = y
8	6, 7	nfbi 1852	. . . 4 |- F/ z ( ph <-> x = y )
9	8	nfal 1860	. . 3 |- F/ z A. x ( ph <-> x = y )
10		equequ2 1694	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
11	10	bibi2d 310	. . . 4 |- ( z = y -> ( ( ph <-> x = z ) <-> ( ph <-> x = y ) ) )
12	11	albidv 1632	. . 3 |- ( z = y -> ( A. x ( ph <-> x = z ) <-> A. x ( ph <-> x = y ) ) )
13	5, 9, 12	cbvex 2038	. 2 |- ( E. z A. x ( ph <-> x = z ) <-> E. y A. x ( ph <-> x = y ) )
14	1, 13	bitri 241	1 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )

Syntax hints:   <-> wb 177   A.wal 1546   E.wex 1547   F/wnf 1550   E!weu 2254

This theorem is referenced by:  eu1  2275  eumo0  2278

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-eu 2258