Theorem euf 1902

Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Hypothesis

Ref	Expression
euf.1	|- (ph -> A.yph)

Assertion

Ref	Expression
euf	|- (E!xph <-> E.yA.x(ph <-> x = y))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem euf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 1900	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
2		euf.1	. . . . 5 |- (ph -> A.yph)
3		ax-17 1416	. . . . 5 |- (x = z -> A.y x = z)
4	2, 3	hbbi 1437	. . . 4 |- ((ph <-> x = z) -> A.y(ph <-> x = z))
5	4	hbal 1363	. . 3 |- (A.x(ph <-> x = z) -> A.yA.x(ph <-> x = z))
6		ax-17 1416	. . 3 |- (A.x(ph <-> x = y) -> A.zA.x(ph <-> x = y))
7		equequ2 1596	. . . . 5 |- (z = y -> (x = z <-> x = y))
8	7	bibi2d 221	. . . 4 |- (z = y -> ((ph <-> x = z) <-> (ph <-> x = y)))
9	8	albidv 1702	. . 3 |- (z = y -> (A.x(ph <-> x = z) <-> A.x(ph <-> x = y)))
10	5, 6, 9	cbvexh 1635	. 2 |- (E.zA.x(ph <-> x = z) <-> E.yA.x(ph <-> x = y))
11	1, 10	bitri 173	1 |- (E!xph <-> E.yA.x(ph <-> x = y))

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240  E.wex 1378  E!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-eu 1900

This theorem is referenced by:  eu1  1922  eumo0  1928