Theorem iotavalb 16394

Description: Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5096.

Assertion

Ref	Expression
iotavalb	|- (E!xph -> (A.x(ph <-> x = y) <-> (iotaxph) = y))

Distinct variable group:   x,y

Proof of Theorem iotavalb

Step	Hyp	Ref	Expression
1		iotaval 5096	. 2 |- (A.x(ph <-> x = y) -> (iotaxph) = y)
2		iotasbc 16383	. . . 4 |- (E!xph -> ([(iotaxph) / z]z = y <-> E.z(A.x(ph <-> x = z) /\ z = y)))
3		iotaexeu 16382	. . . . 5 |- (E!xph -> (iotaxph) e. _V)
4		eqsbc3 2494	. . . . 5 |- ((iotaxph) e. _V -> ([(iotaxph) / z]z = y <-> (iotaxph) = y))
5	3, 4	syl 12	. . . 4 |- (E!xph -> ([(iotaxph) / z]z = y <-> (iotaxph) = y))
6	2, 5	bitr3d 589	. . 3 |- (E!xph -> (E.z(A.x(ph <-> x = z) /\ z = y) <-> (iotaxph) = y))
7		equequ2 1495	. . . . . . 7 |- (z = y -> (x = z <-> x = y))
8	7	bibi2d 680	. . . . . 6 |- (z = y -> ((ph <-> x = z) <-> (ph <-> x = y)))
9	8	albidv 1656	. . . . 5 |- (z = y -> (A.x(ph <-> x = z) <-> A.x(ph <-> x = y)))
10	9	biimpac 462	. . . 4 |- ((A.x(ph <-> x = z) /\ z = y) -> A.x(ph <-> x = y))
11	10	19.23aiv 1674	. . 3 |- (E.z(A.x(ph <-> x = z) /\ z = y) -> A.x(ph <-> x = y))
12	6, 11	syl6bir 232	. 2 |- (E!xph -> ((iotaxph) = y -> A.x(ph <-> x = y)))
13	1, 12	impbid2 576	1 |- (E!xph -> (A.x(ph <-> x = y) <-> (iotaxph) = y))

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:  iotavalsb 16398

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

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