Theorem mo2r 2139

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 1572	. . . 4 |- ( ph -> A. y ph )
3	2	eu3h 2132	. . 3 |- ( E! x ph <-> ( E. x ph /\ E. y A. x ( ph -> x = y ) ) )
4	3	simplbi2com 1494	. 2 |- ( E. y A. x ( ph -> x = y ) -> ( E. x ph -> E! x ph ) )
5		df-mo 2090	. 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 1400   F/wnf 1513   E.wex 1545   E!weu 2086   E*wmo 2087

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by:  mo2icl  3005  rmo2ilem  3142  dffun5r  5384  frecuzrdgtcl  10827  frecuzrdgfunlem  10834