Theorem mobidh 2089

Description: Formula-building rule for "at most one" quantifier (deduction form). (Contributed by NM, 8-Mar-1995.)

Hypotheses

Ref	Expression
mobidh.1	|- ( ph -> A. x ph )
mobidh.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
mobidh	|- ( ph -> ( E* x ps <-> E* x ch ) )

Proof of Theorem mobidh

Step	Hyp	Ref	Expression
1		mobidh.1	. . . 4 |- ( ph -> A. x ph )
2		mobidh.2	. . . 4 |- ( ph -> ( ps <-> ch ) )
3	1, 2	exbidh 1638	. . 3 |- ( ph -> ( E. x ps <-> E. x ch ) )
4	1, 2	eubidh 2061	. . 3 |- ( ph -> ( E! x ps <-> E! x ch ) )
5	3, 4	imbi12d 234	. 2 |- ( ph -> ( ( E. x ps -> E! x ps ) <-> ( E. x ch -> E! x ch ) ) )
6		df-mo 2059	. 2 |- ( E* x ps <-> ( E. x ps -> E! x ps ) )
7		df-mo 2059	. 2 |- ( E* x ch <-> ( E. x ch -> E! x ch ) )
8	5, 6, 7	3bitr4g 223	1 |- ( ph -> ( E* x ps <-> E* x ch ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   E.wex 1516   E!weu 2055   E*wmo 2056

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-eu 2058  df-mo 2059

This theorem is referenced by:  euan  2112