Theorem reu2 2723

Description: A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
reu2	|- (E!x e. A ph <-> (E.x e. A ph /\ A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y)))

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

Allowed substitution hint:   ph(x)

Proof of Theorem reu2

Step	Hyp	Ref	Expression
1		nfv 1418	. . 3 |- F/y(x e. A /\ ph)
2	1	eu2 1941	. 2 |- (E!x(x e. A /\ ph) <-> (E.x(x e. A /\ ph) /\ A.xA.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y)))
3		df-reu 2307	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
4		df-rex 2306	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
5		df-ral 2305	. . . 4 |- (A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y) <-> A.x(x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
6		19.21v 1750	. . . . . 6 |- (A.y(x e. A -> (y e. A -> ((ph /\ [y / x]ph) -> x = y))) <-> (x e. A -> A.y(y e. A -> ((ph /\ [y / x]ph) -> x = y))))
7		nfv 1418	. . . . . . . . . . . . 13 |- F/x y e. A
8		nfs1v 1812	. . . . . . . . . . . . 13 |- F/x[y / x]ph
9	7, 8	nfan 1454	. . . . . . . . . . . 12 |- F/x(y e. A /\ [y / x]ph)
10		eleq1 2097	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
11		sbequ12 1651	. . . . . . . . . . . . 13 |- (x = y -> (ph <-> [y / x]ph))
12	10, 11	anbi12d 442	. . . . . . . . . . . 12 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
13	9, 12	sbie 1671	. . . . . . . . . . 11 |- ([y / x](x e. A /\ ph) <-> (y e. A /\ [y / x]ph))
14	13	anbi2i 430	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)))
15		an4 520	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)) <-> ((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)))
16	14, 15	bitri 173	. . . . . . . . 9 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)))
17	16	imbi1i 227	. . . . . . . 8 |- ((((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> (((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)) -> x = y))
18		impexp 250	. . . . . . . 8 |- ((((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)) -> x = y) <-> ((x e. A /\ y e. A) -> ((ph /\ [y / x]ph) -> x = y)))
19		impexp 250	. . . . . . . 8 |- (((x e. A /\ y e. A) -> ((ph /\ [y / x]ph) -> x = y)) <-> (x e. A -> (y e. A -> ((ph /\ [y / x]ph) -> x = y))))
20	17, 18, 19	3bitri 195	. . . . . . 7 |- ((((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> (x e. A -> (y e. A -> ((ph /\ [y / x]ph) -> x = y))))
21	20	albii 1356	. . . . . 6 |- (A.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> A.y(x e. A -> (y e. A -> ((ph /\ [y / x]ph) -> x = y))))
22		df-ral 2305	. . . . . . 7 |- (A.y e. A ((ph /\ [y / x]ph) -> x = y) <-> A.y(y e. A -> ((ph /\ [y / x]ph) -> x = y)))
23	22	imbi2i 215	. . . . . 6 |- ((x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)) <-> (x e. A -> A.y(y e. A -> ((ph /\ [y / x]ph) -> x = y))))
24	6, 21, 23	3bitr4i 201	. . . . 5 |- (A.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> (x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
25	24	albii 1356	. . . 4 |- (A.xA.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y) <-> A.x(x e. A -> A.y e. A ((ph /\ [y / x]ph) -> x = y)))
26	5, 25	bitr4i 176	. . 3 |- (A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y) <-> A.xA.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y))
27	4, 26	anbi12i 433	. 2 |- ((E.x e. A ph /\ A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y)) <-> (E.x(x e. A /\ ph) /\ A.xA.y(((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) -> x = y)))
28	2, 3, 27	3bitr4i 201	1 |- (E!x e. A ph <-> (E.x e. A ph /\ A.x e. A A.y e. A ((ph /\ [y / x]ph) -> x = y)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240  E.wex 1378   e. wcel 1390  [wsb 1642  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-cleq 2030  df-clel 2033  df-ral 2305  df-rex 2306  df-reu 2307

This theorem is referenced by: (None)