Theorem eubid 1385

Description: Formula-building rule for uniqueness quantifier (deduction rule).

Hypotheses

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

Assertion

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

Proof of Theorem eubid

Step	Hyp	Ref	Expression
1		eubid.1	. . . 4 |- (ph -> A.xph)
2		eubid.2	. . . . 5 |- (ph -> (ps <-> ch))
3	2	bibi1d 619	. . . 4 |- (ph -> ((ps <-> x = y) <-> (ch <-> x = y)))
4	1, 3	albid 1104	. . 3 |- (ph -> (A.x(ps <-> x = y) <-> A.x(ch <-> x = y)))
5	4	exbidv 1279	. 2 |- (ph -> (E.yA.x(ps <-> x = y) <-> E.yA.x(ch <-> x = y)))
6		df-eu 1382	. 2 |- (E!xps <-> E.yA.x(ps <-> x = y))
7		df-eu 1382	. 2 |- (E!xch <-> E.yA.x(ch <-> x = y))
8	5, 6, 7	3bitr4g 555	1 |- (ph -> (E!xps <-> E!xch))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954  E.wex 980  E!weu 1380

This theorem is referenced by:  eubidv 1386  eubii 1387  euor 1398  mobid 1404  reueq1f 1785

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-eu 1382

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