Theorem eubid 2211

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

Hypotheses

Ref	Expression
eubid.1	|- F/xph
eubid.2	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
eubid	|- (ph -> (E!xps <-> E!xch))

Proof of Theorem eubid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eubid.1	. . . 4 |- F/xph
2		eubid.2	. . . . 5 |- (ph -> (ps <-> ch))
3	2	bibi1d 310	. . . 4 |- (ph -> ((ps <-> x = y) <-> (ch <-> x = y)))
4	1, 3	albid 1772	. . 3 |- (ph -> (A.x(ps <-> x = y) <-> A.x(ch <-> x = y)))
5	4	exbidv 1626	. 2 |- (ph -> (E.yA.x(ps <-> x = y) <-> E.yA.x(ch <-> x = y)))
6		df-eu 2208	. 2 |- (E!xps <-> E.yA.x(ps <-> x = y))
7		df-eu 2208	. 2 |- (E!xch <-> E.yA.x(ch <-> x = y))
8	5, 6, 7	3bitr4g 279	1 |- (ph -> (E!xps <-> E!xch))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642  E!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  eubidv  2212  euor  2231  mobid  2238  euan  2261  eupickbi  2270  euor2  2272  reubida  2794  reueq1f  2806