Theorem reu2 1338

Description: A way of expressing restricted uniqueness.

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 group(s):   x,y,A   ph,y

Proof of Theorem reu2

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
2	1	eu2 1023	. 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 1207	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
4		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
5		df-ral 1205	. . . 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 942	. . . . . 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		ax-17 925	. . . . . . . . . . . . 13 |- (y e. A -> A.x y e. A)
8		hbs1 986	. . . . . . . . . . . . 13 |- ([y / x]ph -> A.x[y / x]ph)
9	7, 8	hban 704	. . . . . . . . . . . 12 |- ((y e. A /\ [y / x]ph) -> A.x(y e. A /\ [y / x]ph))
10		eleq1 1149	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
11		sbequ12 865	. . . . . . . . . . . . 13 |- (x = y -> (ph <-> [y / x]ph))
12	10, 11	anbi12d 476	. . . . . . . . . . . 12 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
13	9, 12	sbie 904	. . . . . . . . . . 11 |- ([y / x](x e. A /\ ph) <-> (y e. A /\ [y / x]ph))
14	13	anbi2i 367	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)))
15		an4 388	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)) <-> ((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)))
16	14, 15	bitr 151	. . . . . . . . 9 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ y e. A) /\ (ph /\ [y / x]ph)))
17	16	imbi1i 161	. . . . . . . 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 276	. . . . . . . 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 276	. . . . . . . 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	3bitr 155	. . . . . . 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	bial 695	. . . . . 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 1205	. . . . . . 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 160	. . . . . 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	3bitr4 158	. . . . 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	bial 695	. . . 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	bitr4 154	. . 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 369	. 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	3bitr4 158	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 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  [wsb 852  E!weu 1007   e. wcel 1092  A.wral 1201  E.wrex 1202  E!wreu 1203

This theorem is referenced by:  reu4 1340

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207

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