Theorem iota4 5100

Description: Theorem *14.22 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
iota4	|- (E!xph -> [(iotaxph) / x]ph)

Proof of Theorem iota4

Step	Hyp	Ref	Expression
1		df-eu 1775	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
2		bi2 166	. . . . . 6 |- ((ph <-> x = z) -> (x = z -> ph))
3	2	alimi 1338	. . . . 5 |- (A.x(ph <-> x = z) -> A.x(x = z -> ph))
4		sb2 1541	. . . . 5 |- (A.x(x = z -> ph) -> [z / x]ph)
5	3, 4	syl 12	. . . 4 |- (A.x(ph <-> x = z) -> [z / x]ph)
6		iotaval 5096	. . . . . 6 |- (A.x(ph <-> x = z) -> (iotaxph) = z)
7	6	eqcomd 1889	. . . . 5 |- (A.x(ph <-> x = z) -> z = (iotaxph))
8		dfsbcq 2455	. . . . 5 |- (z = (iotaxph) -> ([z / x]ph <-> [(iotaxph) / x]ph))
9	7, 8	syl 12	. . . 4 |- (A.x(ph <-> x = z) -> ([z / x]ph <-> [(iotaxph) / x]ph))
10	5, 9	mpbid 212	. . 3 |- (A.x(ph <-> x = z) -> [(iotaxph) / x]ph)
11	10	19.23aiv 1674	. 2 |- (E.zA.x(ph <-> x = z) -> [(iotaxph) / x]ph)
12	1, 11	sylbi 216	1 |- (E!xph -> [(iotaxph) / x]ph)

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298  E.wex 1326  [wsbc 1534  E!weu 1771  iotacio 5087

This theorem is referenced by:  iota4an 5101  iotacl 5103  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-eu 1775  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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