Theorem hbmo 2125

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 2090	. 2 |- ( E* y ph <-> ( E. y ph -> E! y ph ) )
2		hbmo.1	. . . 4 |- ( ph -> A. x ph )
3	2	hbex 1689	. . 3 |- ( E. y ph -> A. x E. y ph )
4	2	hbeu 2107	. . 3 |- ( E! y ph -> A. x E! y ph )
5	3, 4	hbim 1598	. 2 |- ( ( E. y ph -> E! y ph ) -> A. x ( E. y ph -> E! y ph ) )
6	1, 5	hbxfrbi 1525	1 |- ( E* y ph -> A. x E* y ph )

Syntax hints:   -> wi 4   A.wal 1400   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-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090

This theorem is referenced by:  moexexdc  2171  2moex  2173  2euex  2174  2exeu  2179