Theorem mobid 2238

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

Hypotheses

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

Assertion

Ref	Expression
mobid	|- (ph -> (E*xps <-> E*xch))

Proof of Theorem mobid

Step	Hyp	Ref	Expression
1		mobid.1	. . . 4 |- F/xph
2		mobid.2	. . . 4 |- (ph -> (ps <-> ch))
3	1, 2	exbid 1773	. . 3 |- (ph -> (E.xps <-> E.xch))
4	1, 2	eubid 2211	. . 3 |- (ph -> (E!xps <-> E!xch))
5	3, 4	imbi12d 311	. 2 |- (ph -> ((E.xps -> E!xps) <-> (E.xch -> E!xch)))
6		df-mo 2209	. 2 |- (E*xps <-> (E.xps -> E!xps))
7		df-mo 2209	. 2 |- (E*xch <-> (E.xch -> E!xch))
8	5, 6, 7	3bitr4g 279	1 |- (ph -> (E*xps <-> E*xch))

Syntax hints:   -> wi 4   <-> wb 176  E.wex 1541   F/wnf 1544  E!weu 2204  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-eu 2208  df-mo 2209

This theorem is referenced by:  mobidv  2239  euan  2261  rmobida  2799  rmoeq1f  2807