Theorem eubid 2289

Description: Formula-building rule for uniqueness quantifier (deduction rule). (Contributed by NM, 9-Jul-1994.)

Hypotheses

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

Assertion

Ref	Expression
eubid	|- ( ph -> ( E! x ps <-> E! x ch ) )

Proof of Theorem eubid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eubid.1	. . . 4 |- F/ x ph
2		eubid.2	. . . . 5 |- ( ph -> ( ps <-> ch ) )
3	2	bibi1d 319	. . . 4 |- ( ph -> ( ( ps <-> x = y ) <-> ( ch <-> x = y ) ) )
4	1, 3	albid 1828	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1685	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 2272	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 2272	. 2 |- ( E! x ch <-> E. y A. x ( ch <-> x = y ) )
8	5, 6, 7	3bitr4g 288	1 |- ( ph -> ( E! x ps <-> E! x ch ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1372   E.wex 1591   F/wnf 1594   E!weu 2268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798

This theorem depends on definitions:  df-bi 185  df-ex 1592  df-nf 1595  df-eu 2272

This theorem is referenced by:  mobid  2290  eubidv  2291  euor  2325  euor2  2327  euor2OLD  2328  euan  2346  euanOLD  2347  eupickbiOLD  2362  reubida  3037  reueq1f  3049  eusv2i  4637  reusv2lem3  4643  eubi  30876