Theorem reu2 1930

Description: A way to express 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 groups:   x,y,A   ph,y

Proof of Theorem reu2

Step	Hyp	Ref	Expression
1		ax-17 971	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
2	1	eu2 1396	. 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 1651	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
4		df-rex 1650	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
5		df-ral 1649	. . . 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 1285	. . . . . 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-17 971	. . . . . . . . . . . . 13 |- (y e. A -> A.x y e. A)
8		hbs1 1332	. . . . . . . . . . . . 13 |- ([y / x]ph -> A.x[y / x]ph)
9	7, 8	hban 1009	. . . . . . . . . . . 12 |- ((y e. A /\ [y / x]ph) -> A.x(y e. A /\ [y / x]ph))
10		eleq1 1534	. . . . . . . . . . . . 13 |- (x = y -> (x e. A <-> y e. A))
11		sbequ12 1181	. . . . . . . . . . . . 13 |- (x = y -> (ph <-> [y / x]ph))
12	10, 11	anbi12d 628	. . . . . . . . . . . 12 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ [y / x]ph)))
13	9, 12	sbie 1196	. . . . . . . . . . 11 |- ([y / x](x e. A /\ ph) <-> (y e. A /\ [y / x]ph))
14	13	anbi2i 480	. . . . . . . . . 10 |- (((x e. A /\ ph) /\ [y / x](x e. A /\ ph)) <-> ((x e. A /\ ph) /\ (y e. A /\ [y / x]ph)))
15		an4 506	. . . . . . . . . 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 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 186	. . . . . . . 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 347	. . . . . . . 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 347	. . . . . . . 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 177	. . . . . . 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 999	. . . . . 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 1649	. . . . . . 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 185	. . . . . 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 183	. . . . 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 999	. . . 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 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 482	. 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 183	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 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  E!weu 1380  A.wral 1645  E.wrex 1646  E!wreu 1647

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-reu 1651

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