Theorem eubid 2488

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 333	. . . 4 |- ( ph -> ( ( ps <-> x = y ) <-> ( ch <-> x = y ) ) )
4	1, 3	albid 2090	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1850	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 2474	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 2474	. 2 |- ( E! x ch <-> E. y A. x ( ch <-> x = y ) )
8	5, 6, 7	3bitr4g 303	1 |- ( ph -> ( E! x ps <-> E! x ch ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   E!weu 2470

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

This theorem is referenced by:  mobid  2489  eubidv  2490  euor  2512  euor2  2514  euan  2530  reubida  3124  reueq1f  3136  eusv2i  4863  reusv2lem3  4871  nbusgredgeu0  26270  eubi  38637