Theorem mobid 2049

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

Hypotheses

Ref	Expression
mobid.1	|- F/ x ph
mobid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem mobid

Step	Hyp	Ref	Expression
1		mobid.1	. . . 4 |- F/ x ph
2		mobid.2	. . . 4 |- ( ph -> ( ps <-> ch ) )
3	1, 2	exbid 1604	. . 3 |- ( ph -> ( E. x ps <-> E. x ch ) )
4	1, 2	eubid 2021	. . 3 |- ( ph -> ( E! x ps <-> E! x ch ) )
5	3, 4	imbi12d 233	. 2 |- ( ph -> ( ( E. x ps -> E! x ps ) <-> ( E. x ch -> E! x ch ) ) )
6		df-mo 2018	. 2 |- ( E* x ps <-> ( E. x ps -> E! x ps ) )
7		df-mo 2018	. 2 |- ( E* x ch <-> ( E. x ch -> E! x ch ) )
8	5, 6, 7	3bitr4g 222	1 |- ( ph -> ( E* x ps <-> E* x ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1448   E.wex 1480   E!weu 2014   E*wmo 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-eu 2017  df-mo 2018

This theorem is referenced by:  mobidv  2050  rmobida  2652  rmoeq1f  2660