Theorem mobid 2489

Description: Formula-building rule for "at most one" quantifier (deduction rule). (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 2091	. . 3 |- ( ph -> ( E. x ps <-> E. x ch ) )
4	1, 2	eubid 2488	. . 3 |- ( ph -> ( E! x ps <-> E! x ch ) )
5	3, 4	imbi12d 334	. 2 |- ( ph -> ( ( E. x ps -> E! x ps ) <-> ( E. x ch -> E! x ch ) ) )
6		df-mo 2475	. 2 |- ( E* x ps <-> ( E. x ps -> E! x ps ) )
7		df-mo 2475	. 2 |- ( E* x ch <-> ( E. x ch -> E! x ch ) )
8	5, 6, 7	3bitr4g 303	1 |- ( ph -> ( E* x ps <-> E* x ch ) )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   F/wnf 1708   E!weu 2470   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  mobidv  2491  moanim  2529  rmobida  3129  rmoeq1f  3137  funcnvmptOLD  29467  funcnvmpt  29468