Theorem hbmo 2118

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 2083	. 2 |- ( E* y ph <-> ( E. y ph -> E! y ph ) )
2		hbmo.1	. . . 4 |- ( ph -> A. x ph )
3	2	hbex 1684	. . 3 |- ( E. y ph -> A. x E. y ph )
4	2	hbeu 2100	. . 3 |- ( E! y ph -> A. x E! y ph )
5	3, 4	hbim 1593	. 2 |- ( ( E. y ph -> E! y ph ) -> A. x ( E. y ph -> E! y ph ) )
6	1, 5	hbxfrbi 1520	1 |- ( E* y ph -> A. x E* y ph )

Syntax hints:   -> wi 4   A.wal 1395   E.wex 1540   E!weu 2079   E*wmo 2080

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083

This theorem is referenced by:  moexexdc  2164  2moex  2166  2euex  2167  2exeu  2172