Theorem reu2 3024

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 1619	. . 3 |- F/y(x e. A /\ ph)
2	1	eu2 2229	. 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 2621	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
4		df-rex 2620	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
5		df-ral 2619	. . . 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 1890	. . . . . 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 1619	. . . . . . . . . . . . 13 |- F/x y e. A
8		nfs1v 2106	. . . . . . . . . . . . 13 |- F/x[y / x]ph
9	7, 8	nfan 1824	. . . . . . . . . . . 12 |- F/x(y e. A /\ [y / x]ph)
10		eleq1 2413	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
11		sbequ12 1919	. . . . . . . . . . . . 13 |- (x = y -> (ph <-> [y / x]ph))
12	10, 11	anbi12d 691	. . . . . . . . . . . 12 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
13	9, 12	sbie 2038	. . . . . . . . . . 11 |- ([y / x](x e. A /\ ph) <-> (y e. A /\ [y / x]ph))
14	13	anbi2i 675	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)))
15		an4 797	. . . . . . . . . 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 240	. . . . . . . . 9 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)))
17	16	imbi1i 315	. . . . . . . 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 433	. . . . . . . 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 433	. . . . . . . 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 262	. . . . . . 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 1566	. . . . . 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 2619	. . . . . . 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 303	. . . . . 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 268	. . . . 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 1566	. . . 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 243	. . 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 678	. 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 268	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   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642  [wsb 1648   e. wcel 1710  E!weu 2204  A.wral 2614  E.wrex 2615  E!wreu 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-ral 2619  df-rex 2620  df-reu 2621

This theorem is referenced by: (None)