Theorem eubidh 1947

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

Hypotheses

Ref	Expression
eubidh.1	|- ( ph -> A. x ph )
eubidh.2	|- ( ph -> ( ps <-> ch ) )

Assertion

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

Proof of Theorem eubidh

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eubidh.1	. . . 4 |- ( ph -> A. x ph )
2		eubidh.2	. . . . 5 |- ( ph -> ( ps <-> ch ) )
3	2	bibi1d 231	. . . 4 |- ( ph -> ( ( ps <-> x = y ) <-> ( ch <-> x = y ) ) )
4	1, 3	albidh 1409	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1746	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 1944	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 1944	. 2 |- ( E! x ch <-> E. y A. x ( ch <-> x = y ) )
8	5, 6, 7	3bitr4g 221	1 |- ( ph -> ( E! x ps <-> E! x ch ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   E.wex 1421   E!weu 1941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-eu 1944

This theorem is referenced by:  euor  1967  mobidh  1975  euan  1997  euor2  1999  eupickbi  2023