Theorem hbmo 1980

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 1945	. 2 |- ( E* y ph <-> ( E. y ph -> E! y ph ) )
2		hbmo.1	. . . 4 |- ( ph -> A. x ph )
3	2	hbex 1567	. . 3 |- ( E. y ph -> A. x E. y ph )
4	2	hbeu 1962	. . 3 |- ( E! y ph -> A. x E! y ph )
5	3, 4	hbim 1477	. 2 |- ( ( E. y ph -> E! y ph ) -> A. x ( E. y ph -> E! y ph ) )
6	1, 5	hbxfrbi 1401	1 |- ( E* y ph -> A. x E* y ph )

Syntax hints:   -> wi 4   A.wal 1282   E.wex 1421   E!weu 1941   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  moexexdc  2025  2moex  2027  2euex  2028  2exeu  2033