Theorem eubid 2021

Description: Formula-building rule for unique existential quantifier (deduction form). (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 232	. . . 4 |- ( ph -> ( ( ps <-> x = y ) <-> ( ch <-> x = y ) ) )
4	1, 3	albid 1603	. . 3 |- ( ph -> ( A. x ( ps <-> x = y ) <-> A. x ( ch <-> x = y ) ) )
5	4	exbidv 1813	. 2 |- ( ph -> ( E. y A. x ( ps <-> x = y ) <-> E. y A. x ( ch <-> x = y ) ) )
6		df-eu 2017	. 2 |- ( E! x ps <-> E. y A. x ( ps <-> x = y ) )
7		df-eu 2017	. 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 1341   F/wnf 1448   E.wex 1480   E!weu 2014

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-eu 2017

This theorem is referenced by:  eubidv  2022  mobid  2049  reubida  2647  reueq1f  2659  eusv2i  4433