Theorem reu3 2725

Description: A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)

Assertion

Ref	Expression
reu3	|- (E!x e. A ph <-> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))

Distinct variable groups:   x,y,A   ph,y

Allowed substitution hint:   ph(x)

Proof of Theorem reu3

Step	Hyp	Ref	Expression
1		reurex 2517	. . 3 |- (E!x e. A ph -> E.x e. A ph)
2		reu6 2724	. . . 4 |- (E!x e. A ph <-> E.y e. A A.x e. A (ph <-> x = y))
3		bi1 111	. . . . . 6 |- ((ph <-> x = y) -> (ph -> x = y))
4	3	ralimi 2378	. . . . 5 |- (A.x e. A (ph <-> x = y) -> A.x e. A (ph -> x = y))
5	4	reximi 2410	. . . 4 |- (E.y e. A A.x e. A (ph <-> x = y) -> E.y e. A A.x e. A (ph -> x = y))
6	2, 5	sylbi 114	. . 3 |- (E!x e. A ph -> E.y e. A A.x e. A (ph -> x = y))
7	1, 6	jca 290	. 2 |- (E!x e. A ph -> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))
8		rexex 2362	. . . 4 |- (E.y e. A A.x e. A (ph -> x = y) -> E.yA.x e. A (ph -> x = y))
9	8	anim2i 324	. . 3 |- ((E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)) -> (E.x e. A ph /\ E.yA.x e. A (ph -> x = y)))
10		nfv 1418	. . . . 5 |- F/y(x e. A /\ ph)
11	10	eu3 1943	. . . 4 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ E.yA.x((x e. A /\ ph) -> x = y)))
12		df-reu 2307	. . . 4 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
13		df-rex 2306	. . . . 5 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
14		df-ral 2305	. . . . . . 7 |- (A.x e. A (ph -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
15		impexp 250	. . . . . . . 8 |- (((x e. A /\ ph) -> x = y) <-> (x e. A -> (ph -> x = y)))
16	15	albii 1356	. . . . . . 7 |- (A.x((x e. A /\ ph) -> x = y) <-> A.x(x e. A -> (ph -> x = y)))
17	14, 16	bitr4i 176	. . . . . 6 |- (A.x e. A (ph -> x = y) <-> A.x((x e. A /\ ph) -> x = y))
18	17	exbii 1493	. . . . 5 |- (E.yA.x e. A (ph -> x = y) <-> E.yA.x((x e. A /\ ph) -> x = y))
19	13, 18	anbi12i 433	. . . 4 |- ((E.x e. A ph /\ E.yA.x e. A (ph -> x = y)) <-> (E.x(x e. A /\ ph) /\ E.yA.x((x e. A /\ ph) -> x = y)))
20	11, 12, 19	3bitr4i 201	. . 3 |- (E!x e. A ph <-> (E.x e. A ph /\ E.yA.x e. A (ph -> x = y)))
21	9, 20	sylibr 137	. 2 |- ((E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)) -> E!x e. A ph)
22	7, 21	impbii 117	1 |- (E!x e. A ph <-> (E.x e. A ph /\ E.y e. A A.x e. A (ph -> x = y)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378   e. wcel 1390  E!weu 1897  A.wral 2300  E.wrex 2301  E!wreu 2302

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  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308

This theorem is referenced by:  reu7  2730  bdreu  9310