Theorem iotain 16381

Description: Equivalence between two different forms of iota.

Assertion

Ref	Expression
iotain	|- (E!xph -> |^|{x | ph} = (iotaxph))

Proof of Theorem iotain

Step	Hyp	Ref	Expression
1		df-eu 1775	. 2 |- (E!xph <-> E.yA.x(ph <-> x = y))
2		visset 2295	. . . . 5 |- y e. _V
3	2	intsn 3252	. . . 4 |- |^|{y} = y
4		hba1 1350	. . . . . . 7 |- (A.x(ph <-> x = y) -> A.xA.x(ph <-> x = y))
5		ax4 1318	. . . . . . 7 |- (A.x(ph <-> x = y) -> (ph <-> x = y))
6	4, 5	abbid 2007	. . . . . 6 |- (A.x(ph <-> x = y) -> {x | ph} = {x | x = y})
7		df-sn 3049	. . . . . 6 |- {y} = {x | x = y}
8	6, 7	syl6eqr 1946	. . . . 5 |- (A.x(ph <-> x = y) -> {x | ph} = {y})
9	8	inteqd 3219	. . . 4 |- (A.x(ph <-> x = y) -> |^|{x | ph} = |^|{y})
10		iotaval 5096	. . . 4 |- (A.x(ph <-> x = y) -> (iotaxph) = y)
11	3, 9, 10	3eqtr4a 1954	. . 3 |- (A.x(ph <-> x = y) -> |^|{x | ph} = (iotaxph))
12	11	19.23aiv 1674	. 2 |- (E.yA.x(ph <-> x = y) -> |^|{x | ph} = (iotaxph))
13	1, 12	sylbi 216	1 |- (E!xph -> |^|{x | ph} = (iotaxph))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298  E.wex 1326  E!weu 1771  {cab 1871  {csn 3044  |^|cint 3214  iotacio 5087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-un 2600  df-in 2603  df-sn 3049  df-pr 3050  df-uni 3178  df-int 3215  df-iota 5089

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