Theorem iotasbc 16383

Description: Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define iota in terms of a function of (iotaxph). Their definition differs in that a function of (iotaxph) evaluates to "false" when there isn't a single x that satisfies ph.

Assertion

Ref	Expression
iotasbc	|- (E!xph -> ([(iotaxph) / y]ps <-> E.y(A.x(ph <-> x = y) /\ ps)))

Distinct variable groups:   x,y   ph,y

Proof of Theorem iotasbc

Step	Hyp	Ref	Expression
1		iotaexeu 16382	. . 3 |- (E!xph -> (iotaxph) e. _V)
2		sbc5g 2470	. . 3 |- ((iotaxph) e. _V -> ([(iotaxph) / y]ps <-> E.y(y = (iotaxph) /\ ps)))
3	1, 2	syl 12	. 2 |- (E!xph -> ([(iotaxph) / y]ps <-> E.y(y = (iotaxph) /\ ps)))
4		eueq 2427	. . . . . . 7 |- ((iotaxph) e. _V <-> E!y y = (iotaxph))
5	1, 4	sylib 215	. . . . . 6 |- (E!xph -> E!y y = (iotaxph))
6		df-eu 1775	. . . . . . 7 |- (E!xph <-> E.yA.x(ph <-> x = y))
7		iotaval 5096	. . . . . . . . . 10 |- (A.x(ph <-> x = y) -> (iotaxph) = y)
8	7	eqcomd 1889	. . . . . . . . 9 |- (A.x(ph <-> x = y) -> y = (iotaxph))
9	8	ancri 321	. . . . . . . 8 |- (A.x(ph <-> x = y) -> (y = (iotaxph) /\ A.x(ph <-> x = y)))
10	9	eximi 1387	. . . . . . 7 |- (E.yA.x(ph <-> x = y) -> E.y(y = (iotaxph) /\ A.x(ph <-> x = y)))
11	6, 10	sylbi 216	. . . . . 6 |- (E!xph -> E.y(y = (iotaxph) /\ A.x(ph <-> x = y)))
12		eupick 1834	. . . . . 6 |- ((E!y y = (iotaxph) /\ E.y(y = (iotaxph) /\ A.x(ph <-> x = y))) -> (y = (iotaxph) -> A.x(ph <-> x = y)))
13	5, 11, 12	syl11anc 524	. . . . 5 |- (E!xph -> (y = (iotaxph) -> A.x(ph <-> x = y)))
14	13, 8	impbid1 575	. . . 4 |- (E!xph -> (y = (iotaxph) <-> A.x(ph <-> x = y)))
15	14	anbi1d 679	. . 3 |- (E!xph -> ((y = (iotaxph) /\ ps) <-> (A.x(ph <-> x = y) /\ ps)))
16	15	exbidv 1657	. 2 |- (E!xph -> (E.y(y = (iotaxph) /\ ps) <-> E.y(A.x(ph <-> x = y) /\ ps)))
17	3, 16	bitrd 587	1 |- (E!xph -> ([(iotaxph) / y]ps <-> E.y(A.x(ph <-> x = y) /\ ps)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  E!weu 1771  _Vcvv 2292  iotacio 5087

This theorem is referenced by:  iotasbc2 16384  iotavalb 16394  fvsb 16430

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-eu 1775  df-mo 1776  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

Copyright terms: Public domain