Theorem mo2r 2130

Description: A condition which implies "at most one". (Contributed by Jim Kingdon, 2-Jul-2018.)

Hypothesis

Ref	Expression
mo2r.1	|- F/ y ph

Assertion

Ref	Expression
mo2r	|- ( E. y A. x ( ph -> x = y ) -> E* x ph )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem mo2r

Step	Hyp	Ref	Expression
1		mo2r.1	. . . . 5 |- F/ y ph
2	1	nfri 1565	. . . 4 |- ( ph -> A. y ph )
3	2	eu3h 2123	. . 3 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	3	simplbi2com 1487	. 2 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ph -> E! x ph ) )
5		df-mo 2081	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6	4, 5	sylibr 134	1 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1393   F/wnf 1506   E.wex 1538   E!weu 2077   E*wmo 2078

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  mo2icl  2982  rmo2ilem  3119  dffun5r  5329  frecuzrdgtcl  10629  frecuzrdgfunlem  10636