Theorem hbmo 2053

Description: Bound-variable hypothesis builder for "at most one". (Contributed by NM, 9-Mar-1995.)

Hypothesis

Ref	Expression
hbmo.1	|- ( ph -> A. x ph )

Assertion

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

Proof of Theorem hbmo

Step	Hyp	Ref	Expression
1		df-mo 2018	. 2 |- ( E* y ph <-> ( E. y ph -> E! y ph ) )
2		hbmo.1	. . . 4 |- ( ph -> A. x ph )
3	2	hbex 1624	. . 3 |- ( E. y ph -> A. x E. y ph )
4	2	hbeu 2035	. . 3 |- ( E! y ph -> A. x E! y ph )
5	3, 4	hbim 1533	. 2 |- ( ( E. y ph -> E! y ph ) -> A. x ( E. y ph -> E! y ph ) )
6	1, 5	hbxfrbi 1460	1 |- ( E* y ph -> A. x E* y ph )

Syntax hints:   -> wi 4   A.wal 1341   E.wex 1480   E!weu 2014   E*wmo 2015

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  moexexdc  2098  2moex  2100  2euex  2101  2exeu  2106