Theorem mo2 2479

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 2477	. 2 |- ( E* x ph <-> E. z A. x ( ph -> x = z ) )
2		mo2.1	. . . . 5 |- F/ y ph
3		nfv 1843	. . . . 5 |- F/ y x = z
4	2, 3	nfim 1825	. . . 4 |- F/ y ( ph -> x = z )
5	4	nfal 2153	. . 3 |- F/ y A. x ( ph -> x = z )
6		nfv 1843	. . 3 |- F/ z A. x ( ph -> x = y )
7		equequ2 1953	. . . . 5 |- ( z = y -> ( x = z <-> x = y ) )
8	7	imbi2d 330	. . . 4 |- ( z = y -> ( ( ph -> x = z ) <-> ( ph -> x = y ) ) )
9	8	albidv 1849	. . 3 |- ( z = y -> ( A. x ( ph -> x = z ) <-> A. x ( ph -> x = y ) ) )
10	5, 6, 9	cbvex 2272	. 2 |- ( E. z A. x ( ph -> x = z ) <-> E. y A. x ( ph -> x = y ) )
11	1, 10	bitri 264	1 |- ( E* x ph <-> E. y A. x ( ph -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  mo3  2507  mo  2508  rmo2  3526  nmo  29325  bj-eu3f  32829  bj-mo3OLD  32832  dffun3f  42429