Theorem mo2 2318

Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)

Hypothesis

Ref	Expression
mo2.1	|- F/ y ph

Assertion

Ref	Expression
mo2	|- ( E* x ph <-> E. y A. x ( ph -> x = y ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem mo2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2316	. 2 |- ( E* x ph <-> E. z A. x ( ph -> x = z ) )
2		mo2.1	. . . . 5 |- F/ y ph
3		nfv 1771	. . . . 5 |- F/ y x = z
4	2, 3	nfim 2013	. . . 4 |- F/ y ( ph -> x = z )
5	4	nfal 2040	. . 3 |- F/ y A. x ( ph -> x = z )
6		nfv 1771	. . 3 |- F/ z A. x ( ph -> x = y )
7		equequ2 1878	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
8	7	imbi2d 322	. . . 4 |- ( z = y -> ( ( ph -> x = z ) <-> ( ph -> x = y ) ) )
9	8	albidv 1777	. . 3 |- ( z = y -> ( A. x ( ph -> x = z ) <-> A. x ( ph -> x = y ) ) )
10	5, 6, 9	cbvex 2125	. 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:   -> wi 4   <-> wb 189   A.wal 1452   E.wex 1673   F/wnf 1677   E*wmo 2310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-eu 2313  df-mo 2314

This theorem is referenced by:  mo3  2346  mo  2347  rmo2  3367  nmo  28169  bj-eu3f  31486  bj-mo3OLD  31491