Theorem mo4f 2003

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 1460	. . 3 |- ( ph -> A. y ph )
2	1	mo3h 1996	. 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 1716	. . . . 5 |- ( [ y / x ] ph <-> ps )
6	5	anbi2i 445	. . . 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 1401	. 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 1283   F/wnf 1390   [wsb 1687   E*wmo 1944

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947

This theorem is referenced by:  mo4  2004  mob2  2782  moop2  4035  dffun4f  4969