Theorem iotaval 5096

Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotaval	|- (A.x(ph <-> x = y) -> (iotaxph) = y)

Distinct variable group:   x,y

Proof of Theorem iotaval

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . 7 |- y e. _V
2		sbeqalb 2503	. . . . . . . 8 |- (y e. _V -> ((A.x(ph <-> x = y) /\ A.x(ph <-> x = z)) -> y = z))
3		equcomi 1487	. . . . . . . 8 |- (y = z -> z = y)
4	2, 3	syl6 25	. . . . . . 7 |- (y e. _V -> ((A.x(ph <-> x = y) /\ A.x(ph <-> x = z)) -> z = y))
5	1, 4	ax-mp 7	. . . . . 6 |- ((A.x(ph <-> x = y) /\ A.x(ph <-> x = z)) -> z = y)
6	5	ex 402	. . . . 5 |- (A.x(ph <-> x = y) -> (A.x(ph <-> x = z) -> z = y))
7		equequ2 1495	. . . . . . . . . 10 |- (y = z -> (x = y <-> x = z))
8	7	eqcoms 1887	. . . . . . . . 9 |- (z = y -> (x = y <-> x = z))
9	8	bibi2d 680	. . . . . . . 8 |- (z = y -> ((ph <-> x = y) <-> (ph <-> x = z)))
10	9	biimpd 170	. . . . . . 7 |- (z = y -> ((ph <-> x = y) -> (ph <-> x = z)))
11	10	alimdv 1668	. . . . . 6 |- (z = y -> (A.x(ph <-> x = y) -> A.x(ph <-> x = z)))
12	11	com12 14	. . . . 5 |- (A.x(ph <-> x = y) -> (z = y -> A.x(ph <-> x = z)))
13	6, 12	impbid 574	. . . 4 |- (A.x(ph <-> x = y) -> (A.x(ph <-> x = z) <-> z = y))
14	13	19.21aiv 1664	. . 3 |- (A.x(ph <-> x = y) -> A.z(A.x(ph <-> x = z) <-> z = y))
15		uniabio 5095	. . 3 |- (A.z(A.x(ph <-> x = z) <-> z = y) -> U.{z | A.x(ph <-> x = z)} = y)
16	14, 15	syl 12	. 2 |- (A.x(ph <-> x = y) -> U.{z | A.x(ph <-> x = z)} = y)
17		dfiota2 5090	. 2 |- (iotaxph) = U.{z | A.x(ph <-> x = z)}
18	16, 17	syl5eq 1940	1 |- (A.x(ph <-> x = y) -> (iotaxph) = y)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292  U.cuni 3177  iotacio 5087

This theorem is referenced by:  iotaequ 5097  iotaex 5099  iota4 5100  iotain 16381  iotaexeu 16382  iotasbc 16383  iota2 16393  iotavalb 16394  euunianOLD 16396  pm14.24 16397  sbiota1 16399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-sbc 2454  df-un 2600  df-sn 3049  df-pr 3050  df-uni 3178  df-iota 5089

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