Theorem mo4f 2356

Description: "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)

Hypotheses

Ref	Expression
mo4f.1	|- F/ x ps
mo4f.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
mo4f	|- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Distinct variable groups:   x, y   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)

Proof of Theorem mo4f

Step	Hyp	Ref	Expression
1		nfv 1772	. . 3 |- F/ y ph
2	1	mo3 2347	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
3		mo4f.1	. . . . . 6 |- F/ x ps
4		mo4f.2	. . . . . 6 |- ( x = y -> ( ph <-> ps ) )
5	3, 4	sbie 2248	. . . . 5 |- ( [ y / x ] ph <-> ps )
6	5	anbi2i 705	. . . 4 |- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
7	6	imbi1i 331	. . 3 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
8	7	2albii 1703	. 2 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
9	2, 8	bitri 257	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1453   F/wnf 1678   [wsb 1808   E*wmo 2311

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-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315

This theorem is referenced by:  mo4  2357  bm1.1  2447  mob2  3230  moop2  4713