Theorem eubid 2283

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 1824	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1681	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 2266	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 2266	. 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 1368   E.wex 1587   F/wnf 1590   E!weu 2262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794

This theorem depends on definitions:  df-bi 185  df-ex 1588  df-nf 1591  df-eu 2266

This theorem is referenced by:  mobid  2284  eubidv  2285  euor  2319  euor2  2321  euor2OLD  2322  euan  2340  euanOLD  2341  eupickbiOLD  2356  reubida  3009  reueq1f  3021  eusv2i  4598  reusv2lem3  4604  eubi  29839