Theorem mo2r 2052

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 1500	. . . 4 |- ( ph -> A. y ph )
3	2	eu3h 2045	. . 3 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	3	simplbi2com 1421	. 2 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ph -> E! x ph ) )
5		df-mo 2004	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6	4, 5	sylibr 133	1 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1330   F/wnf 1437   E.wex 1469   E!weu 2000   E*wmo 2001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by:  mo2icl  2867  rmo2ilem  3002  dffun5r  5143  frecuzrdgtcl  10216  frecuzrdgfunlem  10223