Theorem mo4f 2001

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		ax-17 1459	. . 3 |- ( ph -> A. y ph )
2	1	mo3h 1994	. 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 1714	. . . . 5 |- ( [ y / x ] ph <-> ps )
6	5	anbi2i 444	. . . 4 |- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
7	6	imbi1i 236	. . 3 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
8	7	2albii 1400	. 2 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
9	2, 8	bitri 182	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   F/wnf 1389   [wsb 1685   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  mo4  2002  mob2  2772  moop2  4006  dffun4f  4938