Theorem mo2r 1993

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 1452	. . . 4 |- ( ph -> A. y ph )
3	2	eu3h 1986	. . 3 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	3	simplbi2com 1373	. 2 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ph -> E! x ph ) )
5		df-mo 1945	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6	4, 5	sylibr 132	1 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1282   F/wnf 1389   E.wex 1421   E!weu 1941   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:  mo2icl  2771  rmo2ilem  2903  dffun5r  4934  frecuzrdgfn  9414