Theorem mo4f 2115

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 1550	. . 3 |- ( ph -> A. y ph )
2	1	mo3h 2108	. 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 1815	. . . . 5 |- ( [ y / x ] ph <-> ps )
6	5	anbi2i 457	. . . 4 |- ( ( ph /\ [ y / x ] ph ) <-> ( ph /\ ps ) )
7	6	imbi1i 238	. . 3 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( ph /\ ps ) -> x = y ) )
8	7	2albii 1495	. 2 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )
9	2, 8	bitri 184	1 |- ( E* x ph <-> A. x A. y ( ( ph /\ ps ) -> x = y ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   F/wnf 1484   [wsb 1786   E*wmo 2056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059

This theorem is referenced by:  mo4  2116  mob2  2957  moop2  4304  dffun4f  5296