Theorem eubid 2337

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 326	. . . 4 |- ( ph -> ( ( ps <-> x = y ) <-> ( ch <-> x = y ) ) )
4	1, 3	albid 1983	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1776	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 2323	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 2323	. 2 |- ( E! x ch <-> E. y A. x ( ch <-> x = y ) )
8	5, 6, 7	3bitr4g 296	1 |- ( ph -> ( E! x ps <-> E! x ch ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450   E.wex 1671   F/wnf 1675   E!weu 2319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950

This theorem depends on definitions:  df-bi 190  df-ex 1672  df-nf 1676  df-eu 2323

This theorem is referenced by:  mobid  2338  eubidv  2339  euor  2361  euor2  2363  euan  2379  reubida  2959  reueq1f  2971  eusv2i  4598  reusv2lem3  4604  eubi  36857  nbusgredgeu0  39606