Theorem mobidh 1934

Description: Formula-building rule for "at most one" quantifier (deduction rule). (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 1505	. . 3 |- ( ph -> ( E. x ps <-> E. x ch ) )
4	1, 2	eubidh 1906	. . 3 |- ( ph -> ( E! x ps <-> E! x ch ) )
5	3, 4	imbi12d 223	. 2 |- ( ph -> ( ( E. x ps -> E! x ps ) <-> ( E. x ch -> E! x ch ) ) )
6		df-mo 1904	. 2 |- ( E* x ps <-> ( E. x ps -> E! x ps ) )
7		df-mo 1904	. 2 |- ( E* x ch <-> ( E. x ch -> E! x ch ) )
8	5, 6, 7	3bitr4g 212	1 |- ( ph -> ( E* x ps <-> E* x ch ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   E.wex 1381   E!weu 1900   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-eu 1903  df-mo 1904

This theorem is referenced by:  euan  1956