Theorem eu3h 1942

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eu3h.1	|- (ph -> A.yph)

Assertion

Ref	Expression
eu3h	|- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y)

Proof of Theorem eu3h

Step	Hyp	Ref	Expression
1		euex 1927	. . 3 |- (E!xph -> E.xph)
2		eu3h.1	. . . 4 |- (ph -> A.yph)
3	2	eumo0 1928	. . 3 |- (E!xph -> E.yA.x(ph -> x = y))
4	1, 3	jca 290	. 2 |- (E!xph -> (E.xph /\ E.yA.x(ph -> x = y)))
5	2	nfi 1348	. . . . 5 |- F/yph
6	5	mo23 1938	. . . 4 |- (E.yA.x(ph -> x = y) -> A.xA.y((ph /\ [y / x]ph) -> x = y))
7	6	anim2i 324	. . 3 |- ((E.xph /\ E.yA.x(ph -> x = y)) -> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
8	5	eu2 1941	. . 3 |- (E!xph <-> (E.xph /\ A.xA.y((ph /\ [y / x]ph) -> x = y)))
9	7, 8	sylibr 137	. 2 |- ((E.xph /\ E.yA.x(ph -> x = y)) -> E!xph)
10	4, 9	impbii 117	1 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378  [wsb 1642  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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  eu3  1943  mo2r  1949  2eu4  1990