Theorem eubidh 2025

Description: Formula-building rule for unique existential quantifier (deduction form). (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 232	. . . 4 |- ( ph -> ( ( ps <-> x = y ) <-> ( ch <-> x = y ) ) )
4	1, 3	albidh 1473	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1818	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 2022	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 2022	. 2 |- ( E! x ch <-> E. y A. x ( ch <-> x = y ) )
8	5, 6, 7	3bitr4g 222	1 |- ( ph -> ( E! x ps <-> E! x ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   E.wex 1485   E!weu 2019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-eu 2022

This theorem is referenced by:  euor  2045  mobidh  2053  euan  2075  euor2  2077  eupickbi  2101