Theorem mo2r 1952

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 1412	. . . 4 |- ( ph -> A. y ph )
3	2	eu3h 1945	. . 3 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	3	simplbi2com 1333	. 2 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ph -> E! x ph ) )
5		df-mo 1904	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6	4, 5	sylibr 137	1 |- ( E. y A. x ( ph -> x = y ) -> E* x ph )

Syntax hints:   -> wi 4   A.wal 1241   F/wnf 1349   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  mo2icl  2720  rmo2ilem  2847  dffun5r  4914  frecuzrdgfn  9198